cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A000292 Tetrahedral (or triangular pyramidal) numbers: a(n) = C(n+2,3) = n*(n+1)*(n+2)/6.

Original entry on oeis.org

0, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969, 1140, 1330, 1540, 1771, 2024, 2300, 2600, 2925, 3276, 3654, 4060, 4495, 4960, 5456, 5984, 6545, 7140, 7770, 8436, 9139, 9880, 10660, 11480, 12341, 13244, 14190, 15180
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

a(n) is the number of balls in a triangular pyramid in which each edge contains n balls.
One of the 5 Platonic polyhedral (tetrahedral, cube, octahedral, dodecahedral and icosahedral) numbers (cf. A053012).
Also (1/6)*(n^3 + 3*n^2 + 2*n) is the number of ways to color the vertices of a triangle using <= n colors, allowing rotations and reflections. Group is the dihedral group D_6 with cycle index (x1^3 + 2*x3 + 3*x1*x2)/6.
Also the convolution of the natural numbers with themselves. - Felix Goldberg (felixg(AT)tx.technion.ac.il), Feb 01 2001
Connected with the Eulerian numbers (1, 4, 1) via 1*a(n-2) + 4*a(n-1) + 1*a(n) = n^3. - Gottfried Helms, Apr 15 2002
a(n) is sum of all the possible products p*q where (p,q) are ordered pairs and p + q = n + 1. E.g., a(5) = 5 + 8 + 9 + 8 + 5 = 35. - Amarnath Murthy, May 29 2003
Number of labeled graphs on n+3 nodes that are triangles. - Jon Perry, Jun 14 2003
Number of permutations of n+3 which have exactly 1 descent and avoid the pattern 1324. - Mike Zabrocki, Nov 05 2004
Schlaefli symbol for this polyhedron: {3,3}.
Transform of n^2 under the Riordan array (1/(1-x^2), x). - Paul Barry, Apr 16 2005
a(n) is a perfect square only for n = {1, 2, 48}. E.g., a(48) = 19600 = 140^2. - Alexander Adamchuk, Nov 24 2006
a(n+1) is the number of terms in the expansion of (a_1 + a_2 + a_3 + a_4)^n. - Sergio Falcon, Feb 12 2007 [Corrected by Graeme McRae, Aug 28 2007]
a(n+1) is the number of terms in the complete homogeneous symmetric polynomial of degree n in 3 variables. - Richard Barnes, Sep 06 2017
This is also the average "permutation entropy", sum((pi(n)-n)^2)/n!, over the set of all possible n! permutations pi. - Jeff Boscole (jazzerciser(AT)hotmail.com), Mar 20 2007
a(n) = (d/dx)(S(n, x), x)|A049310.%20-%20_Wolfdieter%20Lang">{x = 2}. First derivative of Chebyshev S-polynomials evaluated at x = 2. See A049310. - _Wolfdieter Lang, Apr 04 2007
If X is an n-set and Y a fixed (n-1)-subset of X then a(n-2) is equal to the number of 3-subsets of X intersecting Y. - Milan Janjic, Aug 15 2007
Complement of A145397; A023533(a(n))=1; A014306(a(n))=0. - Reinhard Zumkeller, Oct 14 2008
Equals row sums of triangle A152205. - Gary W. Adamson, Nov 29 2008
a(n) is the number of gifts received from the lyricist's true love up to and including day n in the song "The Twelve Days of Christmas". a(12) = 364, almost the number of days in the year. - Bernard Hill (bernard(AT)braeburn.co.uk), Dec 05 2008
Sequence of the absolute values of the z^1 coefficients of the polynomials in the GF2 denominators of A156925. See A157703 for background information. - Johannes W. Meijer, Mar 07 2009
Starting with 1 = row sums of triangle A158823. - Gary W. Adamson, Mar 28 2009
Wiener index of the path with n edges. - Eric W. Weisstein, Apr 30 2009
This is a 'Matryoshka doll' sequence with alpha=0, the multiplicative counterpart is A000178: seq(add(add(i,i=alpha..k),k=alpha..n),n=alpha..50). - Peter Luschny, Jul 14 2009
a(n) is the number of nondecreasing triples of numbers from a set of size n, and it is the number of strictly increasing triples of numbers from a set of size n+2. - Samuel Savitz, Sep 12 2009 [Corrected and enhanced by Markus Sigg, Sep 24 2023]
a(n) is the number of ordered sequences of 4 nonnegative integers that sum to n. E.g., a(2) = 10 because 2 = 2 + 0 + 0 + 0 = 1 + 1 + 0 + 0 = 0 + 2 + 0 + 0 = 1 + 0 + 1 + 0 = 0 + 1 + 1 + 0 = 0 + 0 + 2 + 0 = 1 + 0 + 0 + 1 = 0 + 1 + 0 + 1 = 0 + 0 + 1 + 1 = 0 + 0 + 0 + 2. - Artur Jasinski, Nov 30 2009
a(n) corresponds to the total number of steps to memorize n verses by the technique described in A173964. - Ibrahima Faye (ifaye2001(AT)yahoo.fr), Feb 22 2010
The number of (n+2)-bit numbers which contain two runs of 1's in their binary expansion. - Vladimir Shevelev, Jul 30 2010
a(n) is also, starting at the second term, the number of triangles formed in n-gons by intersecting diagonals with three diagonal endpoints (see the first column of the table in Sommars link). - Alexandre Wajnberg, Aug 21 2010
Column sums of:
1 4 9 16 25...
1 4 9...
1...
..............
--------------
1 4 10 20 35...
From Johannes W. Meijer, May 20 2011: (Start)
The Ca3, Ca4, Gi3 and Gi4 triangle sums (see A180662 for their definitions) of the Connell-Pol triangle A159797 are linear sums of shifted versions of the duplicated tetrahedral numbers, e.g., Gi3(n) = 17*a(n) + 19*a(n-1) and Gi4(n) = 5*a(n) + a(n-1).
Furthermore the Kn3, Kn4, Ca3, Ca4, Gi3 and Gi4 triangle sums of the Connell sequence A001614 as a triangle are also linear sums of shifted versions of the sequence given above. (End)
a(n-2)=N_0(n), n >= 1, with a(-1):=0, is the number of vertices of n planes in generic position in three-dimensional space. See a comment under A000125 for general arrangement. Comment to Arnold's problem 1990-11, see the Arnold reference, p. 506. - Wolfdieter Lang, May 27 2011
We consider optimal proper vertex colorings of a graph G. Assume that the labeling, i.e., coloring starts with 1. By optimality we mean that the maximum label used is the minimum of the maximum integer label used across all possible labelings of G. Let S=Sum of the differences |l(v) - l(u)|, the sum being over all edges uv of G and l(w) is the label associated with a vertex w of G. We say G admits unique labeling if all possible labelings of G is S-invariant and yields the same integer partition of S. With an offset this sequence gives the S-values for the complete graph on n vertices, n = 2, 3, ... . - K.V.Iyer, Jul 08 2011
Central term of commutator of transverse Virasoro operators in 4-D case for relativistic quantum open strings (ref. Zwiebach). - Tom Copeland, Sep 13 2011
Appears as a coefficient of a Sturm-Liouville operator in the Ovsienko reference on page 43. - Tom Copeland, Sep 13 2011
For n > 0: a(n) is the number of triples (u,v,w) with 1 <= u <= v <= w <= n, cf. A200737. - Reinhard Zumkeller, Nov 21 2011
Regarding the second comment above by Amarnath Murthy (May 29 2003), see A181118 which gives the sequence of ordered pairs. - L. Edson Jeffery, Dec 17 2011
The dimension of the space spanned by the 3-form v[ijk] that couples to M2-brane worldsheets wrapping 3-cycles inside tori (ref. Green, Miller, Vanhove eq. 3.9). - Stephen Crowley, Jan 05 2012
a(n+1) is the number of 2 X 2 matrices with all terms in {0, 1, ..., n} and (sum of terms) = n. Also, a(n+1) is the number of 2 X 2 matrices with all terms in {0, 1, ..., n} and (sum of terms) = 3*n. - Clark Kimberling, Mar 19 2012
Using n + 4 consecutive triangular numbers t(1), t(2), ..., t(n+4), where n is the n-th term of this sequence, create a polygon by connecting points (t(1), t(2)) to (t(2), t(3)), (t(2), t(3)) to (t(3), t(4)), ..., (t(1), t(2)) to (t(n+3), t(n+4)). The area of this polygon will be one-half of each term in this sequence. - J. M. Bergot, May 05 2012
Pisano period lengths: 1, 4, 9, 8, 5, 36, 7, 16, 27, 20, 11, 72, 13, 28, 45, 32, 17,108, 19, 40, ... . (The Pisano sequence modulo m is the auxiliary sequence p(n) = a(n) mod m, n >= 1, for some m. p(n) is periodic for all sequences with rational g.f., like this one, and others. The lengths of the period of p(n) are quoted here for m>=1.) - R. J. Mathar, Aug 10 2012
a(n) is the maximum possible number of rooted triples consistent with any phylogenetic tree (level-0 phylogenetic network) containing exactly n+2 leaves. - Jesper Jansson, Sep 10 2012
For n > 0, the digital roots of this sequence A010888(a(n)) form the purely periodic 27-cycle {1, 4, 1, 2, 8, 2, 3, 3, 3, 4, 7, 4, 5, 2, 5, 6, 6, 6, 7, 1, 7, 8, 5, 8, 9, 9, 9}, which just rephrases the Pisano period length above. - Ant King, Oct 18 2012
a(n) is the number of functions f from {1, 2, 3} to {1, 2, ..., n + 4} such that f(1) + 1 < f(2) and f(2) + 1 < f(3). - Dennis P. Walsh, Nov 27 2012
a(n) is the Szeged index of the path graph with n+1 vertices; see the Diudea et al. reference, p. 155, Eq. (5.8). - Emeric Deutsch, Aug 01 2013
Also the number of permutations of length n that can be sorted by a single block transposition. - Vincent Vatter, Aug 21 2013
From J. M. Bergot, Sep 10 2013: (Start)
a(n) is the 3 X 3 matrix determinant
| C(n,1) C(n,2) C(n,3) |
| C(n+1,1) C(n+1,2) C(n+1,3) |
| C(n+2,1) C(n+2,2) C(n+2,3) |
(End)
In physics, a(n)/2 is the trace of the spin operator S_z^2 for a particle with spin S=n/2. For example, when S=3/2, the S_z eigenvalues are -3/2, -1/2, +1/2, +3/2 and the sum of their squares is 10/2 = a(3)/2. - Stanislav Sykora, Nov 06 2013
a(n+1) = (n+1)*(n+2)*(n+3)/6 is also the dimension of the Hilbert space of homogeneous polynomials of degree n. - L. Edson Jeffery, Dec 12 2013
For n >= 4, a(n-3) is the number of permutations of 1,2...,n with the distribution of up (1) - down (0) elements 0...0111 (n-4 zeros), or, equivalently, a(n-3) is up-down coefficient {n,7} (see comment in A060351). - Vladimir Shevelev, Feb 15 2014
a(n) is one-half the area of the region created by plotting the points (n^2,(n+1)^2). A line connects points (n^2,(n+1)^2) and ((n+1)^2, (n+2)^2) and a line is drawn from (0,1) to each increasing point. From (0,1) to (4,9) the area is 2; from (0,1) to (9,16) the area is 8; further areas are 20,40,70,...,2*a(n). - J. M. Bergot, May 29 2014
Beukers and Top prove that no tetrahedral number > 1 equals a square pyramidal number A000330. - Jonathan Sondow, Jun 21 2014
a(n+1) is for n >= 1 the number of nondecreasing n-letter words over the alphabet [4] = {1, 2, 3, 4} (or any other four distinct numbers). a(2+1) = 10 from the words 11, 22, 33, 44, 12, 13, 14, 23, 24, 34; which is also the maximal number of distinct elements in a symmetric 4 X 4 matrix. Inspired by the Jul 20 2014 comment by R. J. Cano on A000582. - Wolfdieter Lang, Jul 29 2014
Degree of the q-polynomial counting the orbits of plane partitions under the action of the symmetric group S3. Orbit-counting generating function is Product_{i <= j <= k <= n} ( (1 - q^(i + j + k - 1))/(1 - q^(i + j + k - 2)) ). See q-TSPP reference. - Olivier Gérard, Feb 25 2015
Row lengths of tables A248141 and A248147. - Reinhard Zumkeller, Oct 02 2014
If n is even then a(n) = Sum_{k=1..n/2} (2k)^2. If n is odd then a(n) = Sum_{k=0..(n-1)/2} (1+2k)^2. This can be illustrated as stacking boxes inside a square pyramid on plateaus of edge lengths 2k or 2k+1, respectively. The largest k are the 2k X 2k or (2k+1) X (2k+1) base. - R. K. Guy, Feb 26 2015
Draw n lines in general position in the plane. Any three define a triangle, so in all we see C(n,3) = a(n-2) triangles (6 lines produce 4 triangles, and so on). - Terry Stickels, Jul 21 2015
a(n-2) = fallfac(n,3)/3!, n >= 3, is also the number of independent components of an antisymmetric tensor of rank 3 and dimension n. Here fallfac is the falling factorial. - Wolfdieter Lang, Dec 10 2015
Number of compositions (ordered partitions) of n+3 into exactly 4 parts. - Juergen Will, Jan 02 2016
Number of weak compositions (ordered weak partitions) of n-1 into exactly 4 parts. - Juergen Will, Jan 02 2016
For n >= 2 gives the number of multiplications of two nonzero matrix elements in calculating the product of two upper n X n triangular matrices. - John M. Coffey, Jun 23 2016
Terms a(4n+1), n >= 0, are odd, all others are even. The 2-adic valuation of the subsequence of every other term, a(2n+1), n >= 0, yields the ruler sequence A007814. Sequence A275019 gives the 2-adic valuation of a(n). - M. F. Hasler, Dec 05 2016
Does not satisfy Benford's law [Ross, 2012]. - N. J. A. Sloane, Feb 12 2017
C(n+2,3) is the number of ways to select 1 triple among n+2 objects, thus a(n) is the coefficient of x1^(n-1)*x3 in exponential Bell polynomial B_{n+2}(x1,x2,...), hence its link with A050534 and A001296 (see formula). - Cyril Damamme, Feb 26 2018
a(n) is also the number of 3-cycles in the (n+4)-path complement graph. - Eric W. Weisstein, Apr 11 2018
a(n) is the general number of all geodetic graphs of diameter n homeomorphic to a complete graph K4. - Carlos Enrique Frasser, May 24 2018
a(n) + 4*a(n-1) + a(n-2) = n^3 = A000578(n), for n >= 0 (extending the a(n) formula given in the name). This is the Worpitzky identity for cubes. (Number of components of the decomposition of a rank 3 tensor in dimension n >= 1 into symmetric, mixed and antisymmetric parts). For a(n-2) see my Dec 10 2015 comment. - Wolfdieter Lang, Jul 16 2019
a(n) also gives the total number of regular triangles of length k (in some length unit), with k from {1, 2, ..., n}, in the matchstick arrangement with enclosing triangle of length n, but only triangles with the orientation of the enclosing triangle are counted. Row sums of unsigned A122432(n-1, k-1), for n >= 1. See the Andrew Howroyd comment in A085691. - Wolfdieter Lang, Apr 06 2020
a(n) is the number of bigrassmannian permutations on n+1 elements, i.e., permutations which have a unique left descent, and a unique right descent. - Rafael Mrden, Aug 21 2020
a(n-2) is the number of chiral pairs of colorings of the edges or vertices of a triangle using n or fewer colors. - Robert A. Russell, Oct 20 2020
a(n-2) is the number of subsets of {1,2,...,n} whose diameters are their size. For example, for n=4, a(2)=4 and the sets are {1,3}, {2,4}, {1,2,4}, {1,3,4}. - Enrique Navarrete, Dec 26 2020
For n>1, a(n-2) is the number of subsets of {1,2,...,n} in which the second largest element is the size of the subset. For example, for n=4, a(2)=4 and the sets are {2,3}, {2,4}, {1,3,4}, {2,3,4}. - Enrique Navarrete, Jan 02 2021
a(n) is the number of binary strings of length n+2 with exactly three 0's. - Enrique Navarrete, Jan 15 2021
From Tom Copeland, Jun 07 2021: (Start)
Aside from the zero, this sequence is the fourth diagonal of the Pascal matrix A007318 and the only nonvanishing diagonal (fourth) of the matrix representation IM = (A132440)^3/3! of the differential operator D^3/3!, when acting on the row vector of coefficients of an o.g.f., or power series.
M = e^{IM} is the lower triangular matrix of coefficients of the Appell polynomial sequence p_n(x) = e^{D^3/3!} x^n = e^{b. D} x^n = (b. + x)^n = Sum_{k=0..n} binomial(n,k) b_n x^{n-k}, where the (b.)^n = b_n have the e.g.f. e^{b.t} = e^{t^3/3!}, which is that for A025035 aerated with double zeros, the first column of M.
See A099174 and A000332 for analogous relationships for the third and fifth diagonals of the Pascal matrix. (End)
a(n) is the number of circles with a radius of integer length >= 1 and center at a grid point in an n X n grid. - Albert Swafford, Jun 11 2021
Maximum Wiener index over all connected graphs with n+1 vertices. - Allan Bickle, Jul 09 2022
The third Euler row (1,4,1) has an additional connection with the tetrahedral numbers besides the n^3 identity stated above: a^2(n) + 4*a^2(n+1) + a^2(n+2) = a(n^2+4n+4), which can be shown with algebra. E.g., a^2(2) + 4*a^2(3) + a^2(4) = 16 + 400 + 400 = a(16). Although an analogous thing happens with the (1,1) row of Euler's triangle and triangular numbers C(n+1,2) = A000217(n) = T(n), namely both T(n-1) + T(n) = n^2 and T^2(n-1) + T^2(n) = T(n^2) are true, only one (the usual identity) still holds for the Euler row (1,11,11,1) and the C(n,4) numbers in A000332. That is, the dot product of (1,11,11,1) with the squares of 4 consecutive terms of A000332 is not generally a term of A000332. - Richard Peterson, Aug 21 2022
For n > 1, a(n-2) is the number of solutions of the Diophantine equation x1 + x2 + x3 + x4 + x5 = n, subject to the constraints 0 <= x1, 1 <= x2, 2 <= x3, 0 <= x4 <= 1, 0 <= x5 and x5 is even. - Daniel Checa, Nov 03 2022
a(n+1) is also the number of vertices of the generalized Pitman-Stanley polytope with parameters 2, n, and vector (1,1, ... ,1), which is integrally equivalent to a flow polytope over the grid graph having 2 rows and n columns. - William T. Dugan, Sep 18 2023
a(n) is the number of binary words of length (n+1) containing exactly one substring 01. a(2) = 4: 001, 010, 011, 101. - Nordine Fahssi, Dec 09 2024
a(n) is the number of directed bishop moves on an n X n chessboard, identified under rotations (0, 90, 180 and 270 degree) and all reflections. - Hilko Koning, Aug 27 2025

Examples

			a(2) = 3*4*5/6 = 10, the number of balls in a pyramid of 3 layers of balls, 6 in a triangle at the bottom, 3 in the middle layer and 1 on top.
Consider the square array
  1  2  3  4  5  6 ...
  2  4  6  8 10 12 ...
  3  6  9 12 16 20 ...
  4  8 12 16 20 24 ...
  5 10 15 20 25 30 ...
  ...
then a(n) = sum of n-th antidiagonal. - _Amarnath Murthy_, Apr 06 2003
G.f. = x + 4*x^2 + 10*x^3 + 20*x^4 + 35*x^5 + 56*x^6 + 84*x^7 + 120*x^8 + 165*x^9 + ...
Example for a(3+1) = 20 nondecreasing 3-letter words over {1,2,3,4}: 111, 222, 333; 444, 112, 113, 114, 223, 224, 122, 224, 133, 233, 144, 244, 344; 123, 124, 134, 234.  4 + 4*3 + 4 = 20. - _Wolfdieter Lang_, Jul 29 2014
Example for a(4-2) = 4 independent components of a rank 3 antisymmetric tensor A of dimension 4: A(1,2,3), A(1,2,4), A(1,3,4) and A(2,3,4). - _Wolfdieter Lang_, Dec 10 2015

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
  • V. I. Arnold (ed.), Arnold's Problems, Springer, 2004, comments on Problem 1990-11 (p. 75), pp. 503-510. Numbers N_0.
  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 194.
  • J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, pp. 44, 70.
  • H. S. M. Coxeter, Polyhedral numbers, pp. 25-35 of R. S. Cohen, J. J. Stachel and M. W. Wartofsky, eds., For Dirk Struik: Scientific, historical and political essays in honor of Dirk J. Struik, Reidel, Dordrecht, 1974.
  • E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93.
  • L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 4.
  • M. V. Diudea, I. Gutman, and J. Lorentz, Molecular Topology, Nova Science, 2001, Huntington, N.Y. pp. 152-156.
  • Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §8.6 Figurate Numbers, pp. 292-293.
  • J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
  • V. Ovsienko and S. Tabachnikov, Projective Differential Geometry Old and New, Cambridge Tracts in Mathematics (no. 165), Cambridge Univ. Press, 2005.
  • Kenneth A Ross, First Digits of Squares and Cubes, Math. Mag. 85 (2012) 36-42. doi:10.4169/math.mag.85.1.36.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • A. Szenes, The combinatorics of the Verlinde formulas (N.J. Hitchin et al., ed.), in Vector bundles in algebraic geometry, Cambridge, 1995.
  • James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 11-13.
  • D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1987, pp. 126-127.
  • B. Zwiebach, A First Course in String Theory, Cambridge, 2004; see p. 226.

Links

Crossrefs

Bisections give A000447 and A002492.
Sums of 2 consecutive terms give A000330.
a(3n-3) = A006566(n). A000447(n) = a(2n-2). A002492(n) = a(2n+1).
Column 0 of triangle A094415.
Cf. A000217 (first differences), A001044, (see above example), A061552, A040977, A133111, A133112, A152205, A158823, A156925, A157703, A173964, A058187, A190717, A190718, A100440, A181118, A222716.
Partial sums are A000332. - Jonathan Vos Post, Mar 27 2011
Cf. A216499 (the analogous sequence for level-1 phylogenetic networks).
Cf. A068980 (partitions), A231303 (spin physics).
Cf. similar sequences listed in A237616.
Cf. A104712 (second column, if offset is 2).
Cf. A145397 (non-tetrahedral numbers). - Daniel Forgues, Apr 11 2015
Cf. A127324.
Cf. A007814, A275019 (2-adic valuation).
Cf. A000578 (cubes), A005900 (octahedral numbers), A006566 (dodecahedral numbers), A006564 (icosahedral numbers).
Cf. A002817 (4-cycle count of \bar P_{n+4}), A060446 (5-cycle count of \bar P_{n+3}), A302695 (6-cycle count of \bar P_{n+5})
Row 2 of A325000 (simplex facets and vertices) and A327084 (simplex edges and ridges).
Cf. A085691 (matchsticks), A122432 (unsigned row sums).
Cf. (triangle colorings) A006527 (oriented), A000290 (achiral), A327085 (chiral simplex edges and ridges).
Row 3 of A321791 (cycles of n colors using k or fewer colors).
Cf. A007318, A025035, A099174.
The Wiener indices of powers of paths for k = 1..6 are given in A000292, A002623, A014125, A122046, A122047, and A175724, respectively.

Programs

  • GAP
    a:=n->Binomial(n+2,3);; A000292:=List([0..50],n->a(n)); # Muniru A Asiru, Feb 28 2018
  • Haskell
    a000292 n = n * (n + 1) * (n + 2) `div` 6
a000292_list = scanl1 (+) a000217_list
-- Reinhard Zumkeller, Jun 16 2013, Feb 09 2012, Nov 21 2011
  • Magma
    [n*(n+1)*(n+2)/6: n in [0..50]]; // Wesley Ivan Hurt, Jun 03 2014
  • Maple
    a:=n->n*(n+1)*(n+2)/6; seq(a(n), n=0..50);
A000292 := n->binomial(n+2,3); seq(A000292(n), n=0..50);
isA000292 := proc(n)
    option remember;
    local a,i ;
    for i from iroot(6*n,3)-1 do
        a := A000292(i) ;
        if a > n then
            return false;
        elif a = n then
            return true;
        end if;
    end do:
end proc: # R. J. Mathar, Aug 14 2024
  • Mathematica
    Table[Binomial[n + 2, 3], {n, 0, 20}] (* Zerinvary Lajos, Jan 31 2010 *)
Accumulate[Accumulate[Range[0, 50]]] (* Harvey P. Dale, Dec 10 2011 *)
Table[n (n + 1)(n + 2)/6, {n,0,100}] (* Wesley Ivan Hurt, Sep 25 2013 *)
Nest[Accumulate, Range[0, 50], 2] (* Harvey P. Dale, May 24 2017 *)
Binomial[Range[20] + 1, 3] (* Eric W. Weisstein, Sep 08 2017 *)
LinearRecurrence[{4, -6, 4, -1}, {0, 1, 4, 10}, 20] (* Eric W. Weisstein, Sep 08 2017 *)
CoefficientList[Series[x/(-1 + x)^4, {x, 0, 20}], x] (* Eric W. Weisstein, Sep 08 2017 *)
Table[Range[n].Range[n,1,-1],{n,0,50}] (* Harvey P. Dale, Mar 02 2024 *)
  • Maxima
    A000292(n):=n*(n+1)*(n+2)/6$ makelist(A000292(n),n,0,60); /* Martin Ettl, Oct 24 2012 */
  • PARI
    a(n) = (n) * (n+1) * (n+2) / 6  \\ corrected by Harry J. Smith, Dec 22 2008
  • PARI
    a=vector(10000);a[2]=1;for(i=3,#a,a[i]=a[i-2]+i*i); \\ Stanislav Sykora, Nov 07 2013
  • PARI
    is(n)=my(k=sqrtnint(6*n,3)); k*(k+1)*(k+2)==6*n \\ Charles R Greathouse IV, Dec 13 2016
  • Python
    # Compare A000217.
def A000292():
    x, y, z = 1, 1, 1
    yield 0
    while True:
        yield x
        x, y, z = x + y + z + 1, y + z + 1, z + 1
a = A000292(); print([next(a) for i in range(45)]) # Peter Luschny, Aug 03 2019

Formula

a(n) = C(n+2,3) = n*(n+1)*(n+2)/6 (see the name).
G.f.: x / (1 - x)^4.
a(n) = -a(-4 - n) for all in Z.
a(n) = Sum_{k=0..n} A000217(k) = Sum_{k=1..n} Sum_{j=0..k} j, partial sums of the triangular numbers.
a(2n)= A002492(n). a(2n+1)=A000447(n+1).
a(n) = Sum_{1 <= i <= j <= n} |i - j|. - Amarnath Murthy, Aug 05 2002
a(n) = (n+3)*a(n-1)/n. - Ralf Stephan, Apr 26 2003
Sums of three consecutive terms give A006003. - Ralf Stephan, Apr 26 2003
Determinant of the n X n symmetric Pascal matrix M_(i, j) = C(i+j+2, i). - Benoit Cloitre, Aug 19 2003
The sum of a series constructed by the products of the index and the length of the series (n) minus the index (i): a(n) = sum[i(n-i)]. - Martin Steven McCormick (mathseq(AT)wazer.net), Apr 06 2005
a(n) = Sum_{k=0..floor((n-1)/2)} (n-2k)^2 [offset 0]; a(n+1) = Sum_{k=0..n} k^2*(1-(-1)^(n+k-1))/2 [offset 0]. - Paul Barry, Apr 16 2005
a(n) = -A108299(n+5, 6) = A108299(n+6, 7). - Reinhard Zumkeller, Jun 01 2005
a(n) = -A110555(n+4, 3). - Reinhard Zumkeller, Jul 27 2005
Values of the Verlinde formula for SL_2, with g = 2: a(n) = Sum_{j=1..n-1} n/(2*sin^2(j*Pi/n)). - Simone Severini, Sep 25 2006
a(n-1) = (1/(1!*2!))*Sum_{1 <= x_1, x_2 <= n} |det V(x_1, x_2)| = (1/2)*Sum_{1 <= i,j <= n} |i-j|, where V(x_1, x_2) is the Vandermonde matrix of order 2. Column 2 of A133112. - Peter Bala, Sep 13 2007
Starting with 1 = binomial transform of [1, 3, 3, 1, ...]; e.g., a(4) = 20 = (1, 3, 3, 1) dot (1, 3, 3, 1) = (1 + 9 + 9 + 1). - Gary W. Adamson, Nov 04 2007
a(n) = A006503(n) - A002378(n). - Reinhard Zumkeller, Sep 24 2008
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n >= 4. - Jaume Oliver Lafont, Nov 18 2008
Sum_{n>=1} 1/a(n) = 3/2, case x = 1 in Gradstein-Ryshik 1.513.7. - R. J. Mathar, Jan 27 2009
E.g.f.:((x^3)/6 + x^2 + x)*exp(x). - Geoffrey Critzer, Feb 21 2009
Limit_{n -> oo} A171973(n)/a(n) = sqrt(2)/2. - Reinhard Zumkeller, Jan 20 2010
With offset 1, a(n) = (1/6)*floor(n^5/(n^2 + 1)). - Gary Detlefs, Feb 14 2010
a(n) = Sum_{k = 1..n} k*(n-k+1). - Vladimir Shevelev, Jul 30 2010
a(n) = (3*n^2 + 6*n + 2)/(6*(h(n+2) - h(n-1))), n > 0, where h(n) is the n-th harmonic number. - Gary Detlefs, Jul 01 2011
a(n) = coefficient of x^2 in the Maclaurin expansion of 1 + 1/(x+1) + 1/(x+1)^2 + 1/(x+1)^3 + ... + 1/(x+1)^n. - Francesco Daddi, Aug 02 2011
a(n) = coefficient of x^4 in the Maclaurin expansion of sin(x)*exp((n+1)*x). - Francesco Daddi, Aug 04 2011
a(n) = 2*A002415(n+1)/(n+1). - Tom Copeland, Sep 13 2011
a(n) = A004006(n) - n - 1. - Reinhard Zumkeller, Mar 31 2012
a(n) = (A007531(n) + A027480(n) + A007290(n))/11. - J. M. Bergot, May 28 2012
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 1. - Ant King, Oct 18 2012
G.f.: x*U(0) where U(k) = 1 + 2*x*(k+2)/( 2*k+1 - x*(2*k+1)*(2*k+5)/(x*(2*k+5)+(2*k+2)/U(k+1) )); (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Dec 01 2012
a(n^2 - 1) = (1/2)*(a(n^2 - n - 2) + a(n^2 + n - 2)) and
a(n^2 + n - 2) - a(n^2 - 1) = a(n-1)*(3*n^2 - 2) = 10*A024166(n-1), by Berselli's formula in A222716. - Jonathan Sondow, Mar 04 2013
G.f.: x + 4*x^2/(Q(0)-4*x) where Q(k) = 1 + k*(x+1) + 4*x - x*(k+1)*(k+5)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Mar 14 2013
a(n+1) = det(C(i+3,j+2), 1 <= i,j <= n), where C(n,k) are binomial coefficients. - Mircea Merca, Apr 06 2013
a(n) = a(n-2) + n^2, for n > 1. - Ivan N. Ianakiev, Apr 16 2013
a(2n) = 4*(a(n-1) + a(n)), for n > 0. - Ivan N. Ianakiev, Apr 26 2013
G.f.: x*G(0)/2, where G(k) = 1 + 1/(1 - x/(x + (k+1)/(k+4)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 02 2013
a(n) = n + 2*a(n-1) - a(n-2), with a(0) = a(-1) = 0. - Richard R. Forberg, Jul 11 2013
a(n)*(m+1)^3 + a(m)*(n+1) = a(n*m + n + m), for any nonnegative integers m and n. This is a 3D analog of Euler's theorem about triangular numbers, namely t(n)*(2m+1)^2 + t(m) = t(2nm + n + m), where t(n) is the n-th triangular number. - Ivan N. Ianakiev, Aug 20 2013
Sum_{n>=0} a(n)/(n+1)! = 2*e/3 = 1.8121878856393... . Sum_{n>=1} a(n)/n! = 13*e/6 = 5.88961062832... . - Richard R. Forberg, Dec 25 2013
a(n+1) = A023855(n+1) + A023856(n). - Wesley Ivan Hurt, Sep 24 2013
a(n) = A024916(n) + A076664(n), n >= 1. - Omar E. Pol, Feb 11 2014
a(n) = A212560(n) - A059722(n). - J. M. Bergot, Mar 08 2014
Sum_{n>=1} (-1)^(n + 1)/a(n) = 12*log(2) - 15/2 = 0.8177661667... See A242024, A242023. - Richard R. Forberg, Aug 11 2014
3/(Sum_{n>=m} 1/a(n)) = A002378(m), for m > 0. - Richard R. Forberg, Aug 12 2014
a(n) = Sum_{i=1..n} Sum_{j=i..n} min(i,j). - Enrique Pérez Herrero, Dec 03 2014
Arithmetic mean of Square pyramidal number and Triangular number: a(n) = (A000330(n) + A000217(n))/2. - Luciano Ancora, Mar 14 2015
a(k*n) = a(k)*a(n) + 4*a(k-1)*a(n-1) + a(k-2)*a(n-2). - Robert Israel, Apr 20 2015
Dirichlet g.f.: (zeta(s-3) + 3*zeta(s-2) + 2*zeta(s-1))/6. - Ilya Gutkovskiy, Jul 01 2016
a(n) = A080851(1,n-1) - R. J. Mathar, Jul 28 2016
a(n) = (A000578(n+1) - (n+1) ) / 6. - Zhandos Mambetaliyev, Nov 24 2016
G.f.: x/(1 - x)^4 = (x * r(x) * r(x^2) * r(x^4) * r(x^8) * ...), where r(x) = (1 + x)^4 = (1 + 4x + 6x^2 + 4x^3 + x^4); and x/(1 - x)^4 = (x * r(x) * r(x^3) * r(x^9) * r(x^27) * ...) where r(x) = (1 + x + x^2)^4. - Gary W. Adamson, Jan 23 2017
a(n) = A000332(n+3) - A000332(n+2). - Bruce J. Nicholson, Apr 08 2017
a(n) = A001296(n) - A050534(n+1). - Cyril Damamme, Feb 26 2018
a(n) = Sum_{k=1..n} (-1)^(n-k)*A122432(n-1, k-1), for n >= 1, and a(0) = 0. - Wolfdieter Lang, Apr 06 2020
From Robert A. Russell, Oct 20 2020: (Start)
a(n) = A006527(n) - a(n-2) = (A006527(n) + A000290(n)) / 2 = a(n-2) + A000290(n).
a(n-2) = A006527(n) - a(n) = (A006527(n) - A000290(n)) / 2 = a(n) - A000290(n).
a(n) = 1*C(n,1) + 2*C(n,2) + 1*C(n,3), where the coefficient of C(n,k) is the number of unoriented triangle colorings using exactly k colors.
a(n-2) = 1*C(n,3), where the coefficient of C(n,k) is the number of chiral pairs of triangle colorings using exactly k colors.
a(n-2) = A327085(2,n). (End)
From Amiram Eldar, Jan 25 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = sinh(sqrt(2)*Pi)/(3*sqrt(2)*Pi).
Product_{n>=2} (1 - 1/a(n)) = sqrt(2)*sinh(sqrt(2)*Pi)/(33*Pi). (End)
a(n) = A002623(n-1) + A002623(n-2), for n>1. - Ivan N. Ianakiev, Nov 14 2021

Extensions

Corrected and edited by Daniel Forgues, May 14 2010

A001477 The nonnegative integers.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Although this is a list, and lists normally have offset 1, it seems better to make an exception in this case. - N. J. A. Sloane, Mar 13 2010
The subsequence 0,1,2,3,4 gives the known values of n such that 2^(2^n)+1 is a prime (see A019434, the Fermat primes). - N. J. A. Sloane, Jun 16 2010
Also: The identity map, defined on the set of nonnegative integers. The restriction to the positive integers yields the sequence A000027. - M. F. Hasler, Nov 20 2013
The number of partitions of 2n into exactly 2 parts. - Colin Barker, Mar 22 2015
The number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 8960 or 168.- Philippe A.J.G. Chevalier, Dec 29 2015
Partial sums give A000217. - Omar E. Pol, Jul 26 2018
First differences are A000012 (the "all 1's" sequence). - M. F. Hasler, May 30 2020
See A061579 for the transposed infinite square matrix, or triangle with rows reversed. - M. F. Hasler, Nov 09 2021
This is the unique sequence (a(n)) that satisfies the inequality a(n+1) > a(a(n)) for all n in N. This simple and surprising result comes from the 6th problem proposed by Bulgaria during the second day of the 19th IMO (1977) in Belgrade (see link and reference). - Bernard Schott, Jan 25 2023

Examples

			Triangular view:
   0
   1   2
   3   4   5
   6   7   8   9
  10  11  12  13  14
  15  16  17  18  19  20
  21  22  23  24  25  26  27
  28  29  30  31  32  33  34  35
  36  37  38  39  40  41  42  43  44
  45  46  47  48  49  50  51  52  53  54

References

  • Maurice Protat, Des Olympiades à l'Agrégation, suite vérifiant f(n+1) > f(f(n)), Problème 7, pp. 31-32, Ellipses, Paris 1997.

Links

Crossrefs

Cf. A000027 (n>=1).
Cf. A000012 (first differences).
Partial sums of A057427. - Jeremy Gardiner, Sep 08 2002
Cf. A038608 (alternating signs), A001787 (binomial transform).
Cf. A055112.
Cf. Boustrophedon transforms: A231179, A000737.
Cf. A245422.
Number of orbits of Aut(Z^7) as function of the infinity norm A000579, A154286, A102860, A002412, A045943, A115067, A008586, A008585, A005843, A000217.
When written as an array, the rows/columns are A000217, A000124, A152948, A152950, A145018, A167499, A166136, A167487... and A000096, A034856, A055998, A046691, A052905, A055999... (with appropriate offsets); cf. analogous lists for A000027 in A185787.
Cf. A000290.
Cf. A061579 (transposed matrix / reversed triangle).

Programs

Formula

a(n) = n.
a(0) = 0, a(n) = a(n-1) + 1.
G.f.: x/(1-x)^2.
Multiplicative with a(p^e) = p^e. - David W. Wilson, Aug 01 2001
When seen as array: T(k, n) = n + (k+n)*(k+n+1)/2. Main diagonal is 2*n*(n+1) (A046092), antidiagonal sums are n*(n+1)*(n+2)/2 (A027480). - Ralf Stephan, Oct 17 2004
Dirichlet generating function: zeta(s-1). - Franklin T. Adams-Watters, Sep 11 2005
E.g.f.: x*e^x. - Franklin T. Adams-Watters, Sep 11 2005
a(0)=0, a(1)=1, a(n) = 2*a(n-1) - a(n-2). - Jaume Oliver Lafont, May 07 2008
Alternating partial sums give A001057 = A000217 - 2*(A008794). - Eric Desbiaux, Oct 28 2008
a(n) = 2*A080425(n) + 3*A008611(n-3), n>1. - Eric Desbiaux, Nov 15 2009
a(n) = A007966(n)*A007967(n). - Reinhard Zumkeller, Jun 18 2011
a(n) = Sum_{k>=0} A030308(n,k)*2^k. - Philippe Deléham, Oct 20 2011
a(n) = 2*A028242(n-1) + (-1)^n*A000034(n-1). - R. J. Mathar, Jul 20 2012
a(n+1) = det(C(i+1,j), 1 <= i, j <= n), where C(n,k) are binomial coefficients. - Mircea Merca, Apr 06 2013
a(n-1) = floor(n/e^(1/n)) for n > 0. - Richard R. Forberg, Jun 22 2013
a(n) = A000027(n) for all n>0.
a(n) = floor(cot(1/(n+1))). - Clark Kimberling, Oct 08 2014
a(0)=0, a(n>0) = 2*z(-1)^[( |z|/z + 3 )/2] + ( |z|/z - 1 )/2 for z = A130472(n>0); a 1 to 1 correspondence between integers and naturals. - Adriano Caroli, Mar 29 2015
G.f. as triangle: x*(1 + (x^2 - 5*x + 2)*y + x*(2*x - 1)*y^2)/((1 - x)^3*(1 - x*y)^3). - Stefano Spezia, Jul 22 2025

A000332 Binomial coefficient binomial(n,4) = n*(n-1)*(n-2)*(n-3)/24.

Original entry on oeis.org

0, 0, 0, 0, 1, 5, 15, 35, 70, 126, 210, 330, 495, 715, 1001, 1365, 1820, 2380, 3060, 3876, 4845, 5985, 7315, 8855, 10626, 12650, 14950, 17550, 20475, 23751, 27405, 31465, 35960, 40920, 46376, 52360, 58905, 66045, 73815, 82251, 91390, 101270, 111930, 123410
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of intersection points of diagonals of convex n-gon where no more than two diagonals intersect at any point in the interior.
Also the number of equilateral triangles with vertices in an equilateral triangular array of points with n rows (offset 1), with any orientation. - Ignacio Larrosa Cañestro, Apr 09 2002. [See Les Reid link for proof. - N. J. A. Sloane, Apr 02 2016] [See Peter Kagey link for alternate proof. - Sameer Gauria, Jul 29 2025]
Start from cubane and attach amino acids according to the reaction scheme that describes the reaction between the active sites. See the hyperlink on chemistry. - Robert G. Wilson v, Aug 02 2002
For n>0, a(n) = (-1/8)*(coefficient of x in Zagier's polynomial P_(2n,n)). (Zagier's polynomials are used by PARI/GP for acceleration of alternating or positive series.)
Figurate numbers based on the 4-dimensional regular convex polytope called the regular 4-simplex, pentachoron, 5-cell, pentatope or 4-hypertetrahedron with Schlaefli symbol {3,3,3}. a(n)=((n*(n-1)*(n-2)*(n-3))/4!). - Michael J. Welch (mjw1(AT)ntlworld.com), Apr 01 2004, R. J. Mathar, Jul 07 2009
Maximal number of crossings that can be created by connecting n vertices with straight lines. - Cameron Redsell-Montgomerie (credsell(AT)uoguelph.ca), Jan 30 2007
If X is an n-set and Y a fixed (n-1)-subset of X then a(n) is equal to the number of 4-subsets of X intersecting Y. - Milan Janjic, Aug 15 2007
Product of four consecutive numbers divided by 24. - Artur Jasinski, Dec 02 2007
The only prime in this sequence is 5. - Artur Jasinski, Dec 02 2007
For strings consisting entirely of 0's and 1's, the number of distinct arrangements of four 1's such that 1's are not adjacent. The shortest possible string is 7 characters, of which there is only one solution: 1010101, corresponding to a(5). An eight-character string has 5 solutions, nine has 15, ten has 35 and so on, congruent to A000332. - Gil Broussard, Mar 19 2008
For a(n)>0, a(n) is pentagonal if and only if 3 does not divide n. All terms belong to the generalized pentagonal sequence (A001318). Cf. A000326, A145919, A145920. - Matthew Vandermast, Oct 28 2008
Nonzero terms = row sums of triangle A158824. - Gary W. Adamson, Mar 28 2009
Except for the 4 initial 0's, is equivalent to the partial sums of the tetrahedral numbers A000292. - Jeremy Cahill (jcahill(AT)inbox.com), Apr 15 2009
If the first 3 zeros are disregarded, that is, if one looks at binomial(n+3, 4) with n>=0, then it becomes a 'Matryoshka doll' sequence with alpha=0: seq(add(add(add(i,i=alpha..k),k=alpha..n),n=alpha..m),m=alpha..50). - Peter Luschny, Jul 14 2009
For n>=1, a(n) is the number of n-digit numbers the binary expansion of which contains two runs of 0's. - Vladimir Shevelev, Jul 30 2010
For n>0, a(n) is the number of crossing set partitions of {1,2,..,n} into n-2 blocks. - Peter Luschny, Apr 29 2011
The Kn3, Ca3 and Gi3 triangle sums of A139600 are related to the sequence given above, e.g., Gi3(n) = 2*A000332(n+3) - A000332(n+2) + 7*A000332(n+1). For the definitions of these triangle sums, see A180662. - Johannes W. Meijer, Apr 29 2011
For n > 3, a(n) is the hyper-Wiener index of the path graph on n-2 vertices. - Emeric Deutsch, Feb 15 2012
Except for the four initial zeros, number of all possible tetrahedra of any size, having the same orientation as the original regular tetrahedron, formed when intersecting the latter by planes parallel to its sides and dividing its edges into n equal parts. - V.J. Pohjola, Aug 31 2012
a(n+3) is the number of different ways to color the faces (or the vertices) of a regular tetrahedron with n colors if we count mirror images as the same.
a(n) = fallfac(n,4)/4! is also the number of independent components of an antisymmetric tensor of rank 4 and dimension n >= 1. Here fallfac is the falling factorial. - Wolfdieter Lang, Dec 10 2015
Does not satisfy Benford's law [Ross, 2012] - N. J. A. Sloane, Feb 12 2017
Number of chiral pairs of colorings of the vertices (or faces) of a regular tetrahedron with n available colors. Chiral colorings come in pairs, each the reflection of the other. - Robert A. Russell, Jan 22 2020
From Mircea Dan Rus, Aug 26 2020: (Start)
a(n+3) is the number of lattice rectangles (squares included) in a staircase of order n; this is obtained by stacking n rows of consecutive unit lattice squares, aligned either to the left or to the right, which consist of 1, 2, 3, ..., n squares and which are stacked either in the increasing or in the decreasing order of their lengths. Below, there is a staircase or order 4 which contains a(7) = 35 rectangles. [See the Teofil Bogdan and Mircea Dan Rus link, problem 3, under A004320]
_
||
|||_
|||_|_
|||_|_|
(End)
a(n+4) is the number of strings of length n on an ordered alphabet of 5 letters where the characters in the word are in nondecreasing order. E.g., number of length-2 words is 15: aa,ab,ac,ad,ae,bb,bc,bd,be,cc,cd,ce,dd,de,ee. - Jim Nastos, Jan 18 2021
From Tom Copeland, Jun 07 2021: (Start)
Aside from the zeros, this is the fifth diagonal of the Pascal matrix A007318, the only nonvanishing diagonal (fifth) of the matrix representation IM = (A132440)^4/4! of the differential operator D^4/4!, when acting on the row vector of coefficients of an o.g.f., or power series.
M = e^{IM} is the matrix of coefficients of the Appell sequence p_n(x) = e^{D^4/4!} x^n = e^{b. D} x^n = (b. + x)^n = Sum_{k=0..n} binomial(n,k) b_n x^{n-k}, where the (b.)^n = b_n have the e.g.f. e^{b.t} = e^{t^4/4!}, which is that for A025036 aerated with triple zeros, the first column of M.
See A099174 and A000292 for analogous relationships for the third and fourth diagonals of the Pascal matrix. (End)
For integer m and positive integer r >= 3, the polynomial a(n) + a(n + m) + a(n + 2*m) + ... + a(n + r*m) in n has its zeros on the vertical line Re(n) = (3 - r*m)/2 in the complex plane. - Peter Bala, Jun 02 2024

Examples

			a(5) = 5 from the five independent components of an antisymmetric tensor A of rank 4 and dimension 5, namely A(1,2,3,4), A(1,2,3,5), A(1,2,4,5), A(1,3,4,5) and A(2,3,4,5). See the Dec 10 2015 comment. - _Wolfdieter Lang_, Dec 10 2015

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 196.
  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 74, Problem 8.
  • John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 70.
  • L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 7.
  • Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §8.6 Figurate Numbers, p. 294.
  • J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • Charles W. Trigg, Mathematical Quickies, New York: Dover Publications, Inc., 1985, p. 53, #191.
  • David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 127.

Links

Crossrefs

binomial(n, k): A161680 (k = 2), A000389 (k = 5), A000579 (k = 6), A000580 (k = 7), A000581 (k = 8), A000582 (k = 9).
Cf. A053134, A053126, A075733, A006322, A006325.
Cf. also A000583, A014820, A092181, A092182, A092183.
Cf. A000217, A000292, A007318 (column k = 4).
Cf. A158824.
Cf. A006008 (Number of ways to color the faces (or vertices) of a regular tetrahedron with n colors when mirror images are counted as two).
Cf. A104712 (third column, k=4).
See A269747 for a 3-D analog.
Cf. A006008 (oriented), A006003 (achiral) tetrahedron colorings.
Row 3 of A325000, col. 4 of A007318.
Cf. A000292, A025036, A099174.

Programs

  • GAP
    A000332 := List([1..10^2], n -> Binomial(n, 4)); # Muniru A Asiru, Oct 16 2017
  • Magma
    [Binomial(n,4): n in [0..50]]; // Vincenzo Librandi, Nov 23 2014
  • Maple
    A000332 := n->binomial(n,4); [seq(binomial(n,4), n=0..100)];
  • Mathematica
    Table[ Binomial[n, 4], {n, 0, 45} ] (* corrected by Harvey P. Dale, Aug 22 2011 *)
Table[(n-4)(n-3)(n-2)(n-1)/24, {n, 100}] (* Artur Jasinski, Dec 02 2007 *)
LinearRecurrence[{5,-10,10,-5,1}, {0,0,0,0,1}, 45] (* Harvey P. Dale, Aug 22 2011 *)
CoefficientList[Series[x^4 / (1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Nov 23 2014 *)
  • PARI
    a(n)=binomial(n,4);
  • Python
    # Starts at a(3), i.e. computes n*(n+1)*(n+2)*(n+3)/24
# which is more in line with A000217 and A000292.
def A000332():
    x, y, z, u = 1, 1, 1, 1
    yield 0
    while True:
        yield x
        x, y, z, u = x + y + z + u + 1, y + z + u + 1, z + u + 1, u + 1
a = A000332(); print([next(a) for i in range(41)]) # Peter Luschny, Aug 03 2019
  • Python
    print([n*(n-1)*(n-2)*(n-3)//24 for n in range(50)])
# Gennady Eremin, Feb 06 2022

Formula

a(n) = n*(n-1)*(n-2)*(n-3)/24.
G.f.: x^4/(1-x)^5. - Simon Plouffe in his 1992 dissertation
a(n) = n*a(n-1)/(n-4). - Benoit Cloitre, Apr 26 2003, R. J. Mathar, Jul 07 2009
a(n) = Sum_{k=1..n-3} Sum_{i=1..k} i*(i+1)/2. - Benoit Cloitre, Jun 15 2003
Convolution of natural numbers {1, 2, 3, 4, ...} and A000217, the triangular numbers {1, 3, 6, 10, ...}. - Jon Perry, Jun 25 2003
a(n) = A110555(n+1,4). - Reinhard Zumkeller, Jul 27 2005
a(n+1) = ((n^5-(n-1)^5) - (n^3-(n-1)^3))/24 - (n^5-(n-1)^5-1)/30; a(n) = A006322(n-2)-A006325(n-1). - Xavier Acloque, Oct 20 2003; R. J. Mathar, Jul 07 2009
a(4*n+2) = Pyr(n+4, 4*n+2) where the polygonal pyramidal numbers are defined for integers A>2 and B>=0 by Pyr(A, B) = B-th A-gonal pyramid number = ((A-2)*B^3 + 3*B^2 - (A-5)*B)/6; For all positive integers i and the pentagonal number function P(x) = x*(3*x-1)/2: a(3*i-2) = P(P(i)) and a(3*i-1) = P(P(i) + i); 1 + 24*a(n) = (n^2 + 3*n + 1)^2. - Jonathan Vos Post, Nov 15 2004
First differences of A000389(n). - Alexander Adamchuk, Dec 19 2004
For n > 3, the sum of the first n-2 tetrahedral numbers (A000292). - Martin Steven McCormick (mathseq(AT)wazer.net), Apr 06 2005 [Corrected by Doug Bell, Jun 25 2017]
Starting (1, 5, 15, 35, ...), = binomial transform of [1, 4, 6, 4, 1, 0, 0, 0, ...]. - Gary W. Adamson, Dec 28 2007
Sum_{n>=4} 1/a(n) = 4/3, from the Taylor expansion of (1-x)^3*log(1-x) in the limit x->1. - R. J. Mathar, Jan 27 2009
A034263(n) = (n+1)*a(n+4) - Sum_{i=0..n+3} a(i). Also A132458(n) = a(n)^2 - a(n-1)^2 for n>0. - Bruno Berselli, Dec 29 2010
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5); a(0)=0, a(1)=0, a(2)=0, a(3)=0, a(4)=1. - Harvey P. Dale, Aug 22 2011
a(n) = (binomial(n-1,2)^2 - binomial(n-1,2))/6. - Gary Detlefs, Nov 20 2011
a(n) = Sum_{k=1..n-2} Sum_{i=1..k} i*(n-k-2). - Wesley Ivan Hurt, Sep 25 2013
a(n) = (A000217(A000217(n-2) - 1))/3 = ((((n-2)^2 + (n-2))/2)^2 - (((n-2)^2 + (n-2))/2))/(2*3). - Raphie Frank, Jan 16 2014
Sum_{n>=0} a(n)/n! = e/24. Sum_{n>=3} a(n)/(n-3)! = 73*e/24. See A067764 regarding the second ratio. - Richard R. Forberg, Dec 26 2013
Sum_{n>=4} (-1)^(n+1)/a(n) = 32*log(2) - 64/3 = A242023 = 0.847376444589... . - Richard R. Forberg, Aug 11 2014
4/(Sum_{n>=m} 1/a(n)) = A027480(m-3), for m>=4. - Richard R. Forberg, Aug 12 2014
E.g.f.: x^4*exp(x)/24. - Robert Israel, Nov 23 2014
a(n+3) = C(n,1) + 3*C(n,2) + 3*C(n,3) + C(n,4). Each term indicates the number of ways to use n colors to color a tetrahedron with exactly 1, 2, 3, or 4 colors.
a(n) = A080852(1,n-4). - R. J. Mathar, Jul 28 2016
From Gary W. Adamson, Feb 06 2017: (Start)
G.f.: Starting (1, 5, 14, ...), x/(1-x)^5 can be written
as (x * r(x) * r(x^2) * r(x^4) * r(x^8) * ...) where r(x) = (1+x)^5;
as (x * r(x) * r(x^3) * r(x^9) * r(x^27) * ...) where r(x) = (1+x+x^2)^5;
as (x * r(x) * r(x^4) * r(x^16) * r(x^64) * ...) where r(x) = (1+x+x^2+x^3)^5;
... (as a conjectured infinite set). (End)
From Robert A. Russell, Jan 22 2020: (Start)
a(n) = A006008(n) - a(n+3) = (A006008(n) - A006003(n)) / 2 = a(n+3) - A006003(n).
a(n+3) = A006008(n) - a(n) = (A006008(n) + A006003(n)) / 2 = a(n) + A006003(n).
a(n) = A007318(n,4).
a(n+3) = A325000(3,n). (End)
Product_{n>=5} (1 - 1/a(n)) = cosh(sqrt(15)*Pi/2)/(100*Pi). - Amiram Eldar, Jan 21 2021

Extensions

Some formulas that referred to another offset corrected by R. J. Mathar, Jul 07 2009

A006003 a(n) = n*(n^2 + 1)/2.

Original entry on oeis.org

0, 1, 5, 15, 34, 65, 111, 175, 260, 369, 505, 671, 870, 1105, 1379, 1695, 2056, 2465, 2925, 3439, 4010, 4641, 5335, 6095, 6924, 7825, 8801, 9855, 10990, 12209, 13515, 14911, 16400, 17985, 19669, 21455, 23346, 25345, 27455, 29679, 32020, 34481, 37065, 39775
Offset: 0

Views

Author

N. J. A. Sloane, Simon Plouffe

Keywords

Comments

Write the natural numbers in groups: 1; 2,3; 4,5,6; 7,8,9,10; ... and add the groups. In other words, "sum of the next n natural numbers". - Felice Russo
Number of rhombi in an n X n rhombus, if 'crossformed' rhombi are allowed. - Matti De Craene (Matti.DeCraene(AT)rug.ac.be), May 14 2000
Also the sum of the integers between T(n-1)+1 and T(n), the n-th triangular number (A000217). Sum of n-th row of A000027 regarded as a triangular array.
Unlike the cubes which have a similar definition, it is possible for 2 terms of this sequence to sum to a third. E.g., a(36) + a(37) = 23346 + 25345 = 48691 = a(46). Might be called 2nd-order triangular numbers, thus defining 3rd-order triangular numbers (A027441) as n(n^3+1)/2, etc. - Jon Perry, Jan 14 2004
Also as a(n)=(1/6)*(3*n^3+3*n), n > 0: structured trigonal diamond numbers (vertex structure 4) (cf. A000330 = alternate vertex; A000447 = structured diamonds; A100145 for more on structured numbers). - James A. Record (james.record(AT)gmail.com), Nov 07 2004
The sequence M(n) of magic constants for n X n magic squares (numbered 1 through n^2) from n=3 begins M(n) = 15, 34, 65, 111, 175, 260, ... - Lekraj Beedassy, Apr 16 2005 [comment corrected by Colin Hall, Sep 11 2009]
The sequence Q(n) of magic constants for the n-queens problem in chess begins 0, 0, 0, 0, 34, 65, 111, 175, 260, ... - Paul Muljadi, Aug 23 2005
Alternate terms of A057587. - Jeremy Gardiner, Apr 10 2005
Also partial differences of A063488(n) = (2*n-1)*(n^2-n+2)/2. a(n) = A063488(n) - A063488(n-1) for n>1. - Alexander Adamchuk, Jun 03 2006
In an n X n grid of numbers from 1 to n^2, select -- in any manner -- one number from each row and column. Sum the selected numbers. The sum is independent of the choices and is equal to the n-th term of this sequence. - F.-J. Papp (fjpapp(AT)umich.edu), Jun 06 2006
Nonnegative X values of solutions to the equation (X-Y)^3 - (X+Y) = 0. To find Y values: b(n) = (n^3-n)/2. - Mohamed Bouhamida, May 16 2006
For the equation: m*(X-Y)^k - (X+Y) = 0 with X >= Y, k >= 2 and m is an odd number the X values are given by the sequence defined by a(n) = (m*n^k+n)/2. The Y values are given by the sequence defined by b(n) = (m*n^k-n)/2. - Mohamed Bouhamida, May 16 2006
If X is an n-set and Y a fixed 3-subset of X then a(n-3) is equal to the number of 4-subsets of X intersecting Y. - Milan Janjic, Jul 30 2007
(m*(2n)^k+n, m*(2n)^k-n) solves the Diophantine equation: 2m*(X-Y)^k - (X+Y) = 0 with X >= Y, k >= 2 where m is a positive integer. - Mohamed Bouhamida, Oct 02 2007
Also c^(1/2) in a^(1/2) + b^(1/2) = c^(1/2) such that a^2 + b = c. - Cino Hilliard, Feb 09 2008
a(n) = n*A000217(n) - Sum_{i=0..n-1} A001477(i). - Bruno Berselli, Apr 25 2010
a(n) is the number of triples (w,x,y) having all terms in {0,...,n} such that at least one of these inequalities fails: x+y < w, y+w < x, w+x < y. - Clark Kimberling, Jun 14 2012
Sum of n-th row of the triangle in A209297. - Reinhard Zumkeller, Jan 19 2013
The sequence starting with "1" is the third partial sum of (1, 2, 3, 3, 3, ...). - Gary W. Adamson, Sep 11 2015
a(n) is the largest eigenvalue of the matrix returned by the MATLAB command magic(n) for n > 0. - Altug Alkan, Nov 10 2015
a(n) is the number of triples (x,y,z) having all terms in {1,...,n} such that all these triangle inequalities are satisfied: x+y > z, y+z > x, z+x > y. - Heinz Dabrock, Jun 03 2016
Shares its digital root with the stella octangula numbers (A007588). See A267017. - Peter M. Chema, Aug 28 2016
Can be proved to be the number of nonnegative solutions of a system of three linear Diophantine equations for n >= 0 even: 2*a_{11} + a_{12} + a_{13} = n, 2*a_{22} + a_{12} + a_{23} = n and 2*a_{33} + a_{13} + a_{23} = n. The number of solutions is f(n) = (1/16)*(n+2)*(n^2 + 4n + 8) and a(n) = n*(n^2 + 1)/2 is obtained by remapping n -> 2*n-2. - Kamil Bradler, Oct 11 2016
For n > 0, a(n) coincides with the trace of the matrix formed by writing the numbers 1...n^2 back and forth along the antidiagonals (proved, see A078475 for the examples of matrix). - Stefano Spezia, Aug 07 2018
The trace of an n X n square matrix where the elements are entered on the ascending antidiagonals. The determinant is A069480. - Robert G. Wilson v, Aug 07 2018
Bisections are A317297 and A005917. - Omar E. Pol, Sep 01 2018
Number of achiral colorings of the vertices (or faces) of a regular tetrahedron with n available colors. An achiral coloring is identical to its reflection. - Robert A. Russell, Jan 22 2020
a(n) is the n-th centered triangular pyramidal number. - Lechoslaw Ratajczak, Nov 02 2021
a(n) is the number of words of length n defined on 4 letters {b,c,d,e} that contain one or no b's, one c or two d's, and any number of e's. For example, a(3) = 15 since the words are (number of permutations in parentheses): bce (6), bdd (3), cee (3), and dde (3). - Enrique Navarrete, Jun 21 2025

Examples

			G.f. = x + 5*x^2 + 15*x^3 + 34*x^4 + 65*x^5 + 111*x^6 + 175*x^7 + 260*x^8 + ...
For a(2)=5, the five tetrahedra have faces AAAA, AAAB, AABB, ABBB, and BBBB with colors A and B. - _Robert A. Russell_, Jan 31 2020

References

  • J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 15, p. 5, Ellipses, Paris 2008.
  • F.-J. Papp, Colloquium Talk, Department of Mathematics, University of Michigan-Dearborn, March 6, 2005.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000330, A000537, A066886, A057587, A027480, A002817 (partial sums).
Cf. A000578 (cubes).
Cf. A007742, A005449, A005448, A118465, A226449, A034262, A080992, A267017.
(1/12)*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, this sequence, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.
Antidiagonal sums of array in A000027. Row sums of the triangular view of A000027.
Cf. A063488 (sum of two consecutive terms), A005917 (bisection), A317297 (bisection).
Cf. A105374 / 8.
Cf. A000292, A011863, A001620, A248177.
Tetrahedron colorings: A006008 (oriented), A000332(n+3) (unoriented), A000332 (chiral), A037270 (edges).
Other polyhedron colorings: A337898 (cube faces, octahedron vertices), A337897 (octahedron faces, cube vertices), A337962 (dodecahedron faces, icosahedron vertices), A337960 (icosahedron faces, dodecahedron vertices).
Row 3 of A325001 (simplex vertices and facets) and A337886 (simplex faces and peaks).

Programs

  • GAP
    a_n:=List([0..nmax], n->n*(n^2 + 1)/2); # Stefano Spezia, Aug 12 2018
  • Haskell
    a006003 n = n * (n ^ 2 + 1) `div` 2
a006003_list = scanl (+) 0 a005448_list
-- Reinhard Zumkeller, Jun 20 2013
  • MATLAB
    % Also works with FreeMat.
for(n=0:nmax); tm=n*(n^2 + 1)/2; fprintf('%d\t%0.f\n', n, tm); end
% Stefano Spezia, Aug 12 2018
  • Magma
    [n*(n^2 + 1)/2 : n in [0..50]]; // Wesley Ivan Hurt, Sep 11 2015
  • Magma
    [Binomial(n,3)+Binomial(n-1,3)+Binomial(n-2,3): n in [2..60]]; // Vincenzo Librandi, Sep 12 2015
  • Mathematica
    Table[ n(n^2 + 1)/2, {n, 0, 45}]
LinearRecurrence[{4,-6,4,-1}, {0,1,5,15},50] (* Harvey P. Dale, May 16 2012 *)
CoefficientList[Series[x (1 + x + x^2)/(x - 1)^4, {x, 0, 45}], x] (* Vincenzo Librandi, Sep 12 2015 *)
With[{n=50},Total/@TakeList[Range[(n(n^2+1))/2],Range[0,n]]] (* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Nov 28 2017 *)
  • Maxima
    a(n):=n*(n^2 + 1)/2$ makelist(a(n), n, 0, nmax); /* Stefano Spezia, Aug 12 2018 */
  • PARI
    {a(n) = n * (n^2 + 1) / 2}; /* Michael Somos, Dec 24 2011 */
  • PARI
    concat(0, Vec(x*(1+x+x^2)/(x-1)^4 + O(x^20))) \\ Felix Fröhlich, Oct 11 2016
  • Python
    def A006003(n): return n*(n**2+1)>>1 # Chai Wah Wu, Mar 25 2024

Formula

a(n) = binomial(n+2, 3) + binomial(n+1, 3) + binomial(n, 3). [corrected by Michel Marcus, Jan 22 2020]
G.f.: x*(1+x+x^2)/(x-1)^4. - Floor van Lamoen, Feb 11 2002
Partial sums of A005448. - Jonathan Vos Post, Mar 16 2006
Binomial transform of [1, 4, 6, 3, 0, 0, 0, ...] = (1, 5, 15, 34, 65, ...). - Gary W. Adamson, Aug 10 2007
a(n) = -a(-n) for all n in Z. - Michael Somos, Dec 24 2011
a(n) = Sum_{k = 1..n} A(k-1, k-1-n) where A(i, j) = i^2 + i*j + j^2 + i + j + 1. - Michael Somos, Jan 02 2012
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), with a(0)=0, a(1)=1, a(2)=5, a(3)=15. - Harvey P. Dale, May 16 2012
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 3. - Ant King, Jun 13 2012
a(n) = A000217(n) + n*A000217(n-1). - Bruno Berselli, Jun 07 2013
a(n) = A057145(n+3,n). - Luciano Ancora, Apr 10 2015
E.g.f.: (1/2)*(2*x + 3*x^2 + x^3)*exp(x). - G. C. Greubel, Dec 18 2015; corrected by Ilya Gutkovskiy, Oct 12 2016
a(n) = T(n) + T(n-1) + T(n-2), where T means the tetrahedral numbers, A000292. - Heinz Dabrock, Jun 03 2016
From Ilya Gutkovskiy, Oct 11 2016: (Start)
Convolution of A001477 and A008486.
Convolution of A000217 and A158799.
Sum_{n>=1} 1/a(n) = H(-i) + H(i) = 1.343731971048019675756781..., where H(k) is the harmonic number, i is the imaginary unit. (End)
a(n) = A000578(n) - A135503(n). - Miquel Cerda, Dec 25 2016
Euler transform of length 3 sequence [5, 0, -1]. - Michael Somos, Dec 25 2016
a(n) = A037270(n)/n for n > 0. - Kritsada Moomuang, Dec 15 2018
a(n) = 3*A000292(n-1) + n. - Bruce J. Nicholson, Nov 23 2019
a(n) = A011863(n) - A011863(n-2). - Bruce J. Nicholson, Dec 22 2019
From Robert A. Russell, Jan 22 2020: (Start)
a(n) = C(n,1) + 3*C(n,2) + 3*C(n,3), where the coefficient of C(n,k) is the number of tetrahedron colorings using exactly k colors.
a(n) = C(n+3,4) - C(n,4).
a(n) = 2*A000332(n+3) - A006008(n) = A006008(n) - 2*A000332(n) = A000332(n+3) - A000332(n).
a(n) = A325001(3,n). (End)
From Amiram Eldar, Aug 21 2023: (Start)
Sum_{n>=1} 1/a(n) = 2 * (A248177 + A001620).
Product_{n>=2} (1 - 1/a(n)) = cosh(sqrt(7)*Pi/2)*cosech(Pi)/4.
Product_{n>=1} (1 + 1/a(n)) = cosh(sqrt(7)*Pi/2)*cosech(Pi). (End)

Extensions

Better description from Albert Rich (Albert_Rich(AT)msn.com), Mar 1997

A045943 Triangular matchstick numbers: a(n) = 3*n*(n+1)/2.

Original entry on oeis.org

0, 3, 9, 18, 30, 45, 63, 84, 108, 135, 165, 198, 234, 273, 315, 360, 408, 459, 513, 570, 630, 693, 759, 828, 900, 975, 1053, 1134, 1218, 1305, 1395, 1488, 1584, 1683, 1785, 1890, 1998, 2109, 2223, 2340, 2460, 2583, 2709, 2838, 2970, 3105, 3243, 3384, 3528
Offset: 0

Views

Author

R. K. Guy

Keywords

Comments

Also, 3 times triangular numbers, a(n) = 3*A000217(n).
In the 24-bit RGB color cube, the number of color-lattice-points in r+g+b = n planes at n < 256 equals the triangular numbers. For n = 256, ..., 765 the number of legitimate color partitions is less than A000217(n) because {r,g,b} components cannot exceed 255. For n = 256, ..., 511, the number of non-color partitions are computable with A045943(n-255), while for n = 512, ..., 765, the number of color points in r+g+b planes equals A000217(765-n). - Labos Elemer, Jun 20 2005
If a 3-set Y and an (n-3)-set Z are disjoint subsets of an n-set X then a(n-3) is the number of 3-subsets of X intersecting both Y and Z. - Milan Janjic, Sep 19 2007
a(n) is also the smallest number that may be written both as the sum of n-1 consecutive positive integers and n consecutive positive integers. - Claudio Meller, Oct 08 2010
For n >= 3, a(n) equals 4^(2+n)*Pi^(1 - n) times the coefficient of zeta(3) in the following integral with upper bound Pi/4 and lower bound 0: int x^(n+1) tan x dx. - John M. Campbell, Jul 17 2011
The difference a(n)-a(n-1) = 3*n, for n >= 1. - Stephen Balaban, Jul 25 2011 [Comment clarified by N. J. A. Sloane, Aug 01 2024]
Sequence found by reading the line from 0, in the direction 0, 3, ..., and the same line from 0, in the direction 0, 9, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. This is one of the orthogonal axes of the spiral; the other is A032528. - Omar E. Pol, Sep 08 2011
A005449(a(n)) = A000332(3n + 3) = C(3n + 3, 4), a second pentagonal number of triangular matchstick number index number. Additionally, a(n) - 2n is a pentagonal number (A000326). - Raphie Frank, Dec 31 2012
Sum of the numbers from n to 2n. - Wesley Ivan Hurt, Nov 24 2015
Number of orbits of Aut(Z^7) as function of the infinity norm (n+1) of the representative integer lattice point of the orbit, when the cardinality of the orbit is equal to 5376 or 17920 or 20160. - Philippe A.J.G. Chevalier, Dec 28 2015
Also the number of 4-cycles in the (n+4)-triangular honeycomb acute knight graph. - Eric W. Weisstein, Jul 27 2017
Number of terms less than 10^k, k=0,1,2,3,...: 1, 3, 8, 26, 82, 258, 816, 2582, 8165, 25820, 81650, 258199, 816497, 2581989, 8164966, ... - Muniru A Asiru, Jan 24 2018
Numbers of the form 3*m*(2*m + 1) for m = 0, -1, 1, -2, 2, -3, 3, ... - Bruno Berselli, Feb 26 2018
Partial sums of A008585. - Omar E. Pol, Jun 20 2018
Column 1 of A273464. (Number of ways to select a unit lozenge inside an isosceles triangle of side length n; all vertices on a hexagonal lattice.) - R. J. Mathar, Jul 10 2019
Total number of pips in the n-th suit of a double-n domino set. - Ivan N. Ianakiev, Aug 23 2020

Examples

			From _Stephen Balaban_, Jul 25 2011: (Start)
T(n), the triangular numbers = number of nodes,
a(n-1) = number of edges in the T(n) graph:
       o    (T(1) = 1, a(0) = 0)
       o
      / \   (T(2) = 3, a(1) = 3)
     o - o
       o
      / \
     o - o  (T(3) = 6, a(2) = 9)
    / \ / \
   o - o - o
... [Corrected by _N. J. A. Sloane_, Aug 01 2024] (End)

References

  • Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 543.

Links

Crossrefs

Cf. A005448, A002378, A046092, A051162, A126804, A001318, A032528.
3 times n-gonal numbers: A033428, A062741, A094159, A152773, A152751, A152759, A152767, A153783, A153448, A153875.
The generalized pentagonal numbers b*n+3*n*(n-1)/2, for b = 1 through 12, form sequences A000326, A005449, A045943, A115067, A140090, A140091, A059845, A140672, A140673, A140674, A140675, A151542.
A diagonal of A010027.
Orbits of Aut(Z^7) as function of the infinity norm A000579, A154286, A102860, A002412, A115067, A008585, A005843, A001477, A000217.
Cf. A027480 (partial sums).
Cf. A002378 (3-cycles in triangular honeycomb acute knight graph), A028896 (5-cycles), A152773 (6-cycles).
This sequence: Sum_{k = n..2*n} k.
Cf. A304993: Sum_{k = n..2*n} k*(k+1)/2.
Cf. A050409: Sum_{k = n..2*n} k^2.
Similar sequences are listed in A316466.

Programs

Formula

a(n) is the sum of n+1 integers starting from n, i.e., 1+2, 2+3+4, 3+4+5+6, 4+5+6+7+8, etc. - Jon Perry, Jan 15 2004
a(n) = A126890(n+1,n-1) for n>1. - Reinhard Zumkeller, Dec 30 2006
a(n) + A145919(3*n+3) = 0. - Matthew Vandermast, Oct 28 2008
a(n) = A000217(2*n) - A000217(n-1); A179213(n) <= a(n). - Reinhard Zumkeller, Jul 05 2010
a(n) = a(n-1)+3*n, n>0. - Vincenzo Librandi, Nov 18 2010
G.f.: 3*x/(1-x)^3. - Bruno Berselli, Jan 21 2011
a(n) = A005448(n+1) - 1. - Omar E. Pol, Oct 03 2011
a(n) = A001477(n)+A000290(n)+A000217(n). - J. M. Bergot, Dec 08 2012
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>2. - Wesley Ivan Hurt, Nov 24 2015
a(n) = A027480(n)-A027480(n-1). - Peter M. Chema, Jan 18 2017.
2*a(n)+1 = A003215(n). - Miquel Cerda, Jan 22 2018
a(n) = T(2*n) - T(n-1), where T(n) = A000217(n). In general, T(k)*T(n) = Sum_{i=0..k-1} (-1)^i*T((k-i)*(n-i)). - Charlie Marion, Dec 06 2020
E.g.f.: 3*exp(x)*x*(2 + x)/2. - Stefano Spezia, May 19 2021
From Amiram Eldar, Jan 10 2022: (Start)
Sum_{n>=1} 1/a(n) = 2/3.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(2*log(2)-1)/3. (End)
Product_{n>=1} (1 - 1/a(n)) = -(3/(2*Pi))*cos(sqrt(11/3)*Pi/2). - Amiram Eldar, Feb 21 2023

A003506 Triangle of denominators in Leibniz's Harmonic Triangle a(n,k), n >= 1, 1 <= k <= n.

Original entry on oeis.org

1, 2, 2, 3, 6, 3, 4, 12, 12, 4, 5, 20, 30, 20, 5, 6, 30, 60, 60, 30, 6, 7, 42, 105, 140, 105, 42, 7, 8, 56, 168, 280, 280, 168, 56, 8, 9, 72, 252, 504, 630, 504, 252, 72, 9, 10, 90, 360, 840, 1260, 1260, 840, 360, 90, 10, 11, 110, 495, 1320, 2310, 2772, 2310, 1320, 495, 110, 11
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Array 1/Beta(n,m) read by antidiagonals. - Michael Somos, Feb 05 2004
a(n,3) = A027480(n-2); a(n,4) = A033488(n-3). - Ross La Haye, Feb 13 2004
a(n,k) = total size of all of the elements of the family of k-size subsets of an n-element set. For example, a 2-element set, say, {1,2}, has 3 families of k-size subsets: one with 1 0-size element, one with 2 1-size elements and one with 1 2-size element; respectively, {{}}, {{1},{2}}, {{1,2}}. - Ross La Haye, Dec 31 2006
Second slice along the 1-2-plane in the cube a(m,n,o) = a(m-1,n,o) + a(m,n-1,o) + a(m,n,o-1) with a(1,0,0)=1 and a(m<>1=0,n>=0,0>=o)=0, for which the first slice is Pascal's triangle (slice read by antidiagonals). - Thomas Wieder, Aug 06 2006
Triangle, read by rows, given by [2,-1/2,1/2,0,0,0,0,0,0,...] DELTA [2,-1/2,1/2,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 07 2007
This sequence * [1/1, 1/2, 1/3, ...] = (1, 3, 7, 15, 31, ...). - Gary W. Adamson, Nov 14 2007
n-th row = coefficients of first derivative of corresponding Pascal's triangle row. Example: x^4 + 4x^3 + 6x^2 + 4x + 1 becomes (4, 12, 12, 4). - Gary W. Adamson, Dec 27 2007
From Paul Curtz, Jun 03 2011: (Start)
Consider
1 1/2 1/3 1/4 1/5
-1/2 -1/6 -1/12 -1/20 -1/30
1/3 1/12 1/30 1/60 1/105
-1/4 -1/20 -1/60 -1/140 -1/280
1/5 1/30 1/105 1/280 1/630
This is an autosequence (the inverse binomial transform is the sequence signed) of the second kind: the main diagonal is 2 times the first upper diagonal.
Note that 2, 12, 60, ... = A005430(n+1), Apery numbers = 2*A002457(n). (End)
From Louis Conover (for the 9th grade G1c mathematics class at the Chengdu Confucius International School), Mar 02 2015: (Start)
The i-th order differences of n^-1 appear in the (i+1)th row.
1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, ...
1/2, 1/6, 1/12, 1/20, 1/30, 1/42, 1/56, 1/72, ...
1/3, 1/12, 1/30, 1/60, 1/105, 1/168, 1/252, 1/360, ...
1/4, 1/20, 1/60, 1/140, 1/280, 1/504, 1/840, 1/1320, ...
1/5, 1/30, 1/105, 1/280, 1/630, 1/1260, 1/2310, 1/3960, ...
1/6, 1/42, 1/168, 1/504, 1/1260, 1/2772, 1/5544, 1/12012, ...
(End)
T(n,k) is the number of edges of distance k from a fixed vertex in the n-dimensional hypercube. - Simon Burton, Nov 04 2022

Examples

			The triangle begins:
  1;
  1/2, 1/2;
  1/3, 1/6, 1/3;
  1/4, 1/12, 1/12, 1/4;
  1/5, 1/20, 1/30, 1/20, 1/5;
  ...
The triangle of denominators begins:
   1
   2   2
   3   6   3
   4  12  12    4
   5  20  30   20    5
   6  30  60   60   30    6
   7  42 105  140  105   42    7
   8  56 168  280  280  168   56    8
   9  72 252  504  630  504  252   72   9
  10  90 360  840 1260 1260  840  360  90  10
  11 110 495 1320 2310 2772 2310 1320 495 110 11

References

  • A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, see 130.
  • B. A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8. English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 38.
  • G. Boole, A Treatise On The Calculus of Finite Differences, Dover, 1960, p. 26.
  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 83, Problem 25.
  • M. Elkadi and B. Mourrain, Symbolic-numeric methods for solving polynomial equations and applications, Chap 3. of A. Dickenstein and I. Z. Emiris, eds., Solving Polynomial Equations, Springer, 2005, pp. 126-168. See p. 152.
  • D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 35.

Links

Crossrefs

Cf. A007622, A128064, A164555/A027642, A356546.
Cf. A094305, A121547, A121306, A119800, A002378, A007318.
Row sums are in A001787. Central column is A002457. Half-diagonal is in A090816. A116071, A215652.
Denominators of i-th order differences of n^-1 are given in: (1st) A002378, (2nd) A027480, (3rd) A033488, (4th) A174002, (5th) A253946. - Louis Conover, Mar 02 2015
Columns k >= 1 (offset 1): A000027, A002378, A027480, A033488, A174002, A253946(n+4), ..., with sum of reciprocals: infinity, 1, 1/2, 1/3, 1/4, 1/5, ..., respectively. - Wolfdieter Lang, Jul 20 2022

Programs

  • Haskell
    a003506 n k = a003506_tabl !! (n-1) !! (n-1)
a003506_row n = a003506_tabl !! (n-1)
a003506_tabl = scanl1 (\xs ys ->
   zipWith (+) (zipWith (+) ([0] ++ xs) (xs ++ [0])) ys) a007318_tabl
a003506_list = concat a003506_tabl
-- Reinhard Zumkeller, Nov 14 2013, Nov 17 2011
  • Maple
    with(combstruct):for n from 0 to 11 do seq(m*count(Combination(n), size=m), m = 1 .. n) od; # Zerinvary Lajos, Apr 09 2008
A003506 := (n,k) -> k*binomial(n,k):
seq(print(seq(A003506(n,k),k=1..n)),n=1..7); # Peter Luschny, May 27 2011
  • Mathematica
    L[n_, 1] := 1/n; L[n_, m_] := L[n, m] = L[n - 1, m - 1] - L[n, m - 1]; Take[ Flatten[ Table[ 1 / L[n, m], {n, 1, 12}, {m, 1, n}]], 66]
t[n_, m_] = Gamma[n]/(Gamma[n - m]*Gamma[m]); Table[Table[t[n, m], {m, 1, n - 1}], {n, 2, 12}]; Flatten[%] (* Roger L. Bagula and Gary W. Adamson, Sep 14 2008 *)
Table[k*Binomial[n,k],{n,1,7},{k,1,n}] (* Peter Luschny, May 27 2011 *)
t[n_, k_] := Denominator[n!*k!/(n+k+1)!]; Table[t[n-k, k] , {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 28 2013 *)
  • PARI
    A(i,j)=if(i<1||j<1,0,1/subst(intformal(x^(i-1)*(1-x)^(j-1)),x,1))
  • PARI
    A(i,j)=if(i<1||j<1,0,1/sum(k=0,i-1,(-1)^k*binomial(i-1,k)/(j+k)))
  • PARI
    {T(n, k) = (n + 1 - k) * binomial( n, k - 1)} /* Michael Somos, Feb 06 2011 */
  • SageMath
    T_row = lambda n: (n*(x+1)^(n-1)).list()
for n in (1..10): print(T_row(n)) # Peter Luschny, Feb 04 2017
# Assuming offset 0:
def A003506(n, k):
    return falling_factorial(n+1,n)//(factorial(k)*factorial(n-k))
for n in range(9): print([A003506(n, k) for k in range(n+1)]) # Peter Luschny, Aug 13 2022

Formula

a(n, 1) = 1/n; a(n, k) = a(n-1, k-1) - a(n, k-1) for k > 1.
Considering the integer values (rather than unit fractions): a(n, k) = k*C(n, k) = n*C(n-1, k-1) = a(n, k-1)*a(n-1, k-1)/(a(n, k-1) - a(n-1, k-1)) = a(n-1, k) + a(n-1, k-1)*k/(k-1) = (a(n-1, k) + a(n-1, k-1))*n/(n-1) = k*A007318(n, k) = n*A007318(n-1, k-1). Row sums of integers are n*2^(n-1) = A001787(n); row sums of the unit fractions are A003149(n-1)/A000142(n). - Henry Bottomley, Jul 22 2002
From Vladeta Jovovic, Nov 01 2003: (Start)
G.f.: x*y/(1-x-y*x)^2.
E.g.f.: x*y*exp(x+x*y). (End)
T(n,k) = n*binomial(n-1,k-1) = n*A007318(n-1,k-1). - Philippe Deléham, Aug 04 2006
Binomial transform of A128064(unsigned). - Gary W. Adamson, Aug 29 2007
From Roger L. Bagula and Gary W. Adamson, Sep 14 2008: (Start)
t(n,m) = Gamma(n)/(Gamma(n - m)*Gamma(m)).
f(s,n) = Integral_{x=0..oo} exp(-s*x)*x^n dx = Gamma(n)/s^n; t(n,m) = f(s,n)/(f(s,n-m)*f(s,m)) = Gamma(n)/(Gamma(n - m)*Gamma(m)); the powers of s cancel out. (End)
From Reinhard Zumkeller, Mar 05 2010: (Start)
T(n,5) = T(n,n-4) = A174002(n-4) for n > 4.
T(2*n,n) = T(2*n,n+1) = A005430(n). (End)
T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) - 2*T(n-2,k-1) - T(n-2,k-2), T(1,1) = 1 and, for n > 1, T(n,k) = 0 if k <= 1 or if k > n. - Philippe Deléham, Mar 17 2012
T(n,k) = Sum_{i=1..k} i*binomial(k,i)*binomial(n+1-k,k+1-i). - Mircea Merca, Apr 11 2012
If we include a main diagonal of zeros so that the array is in the form
0
1 0
2 2 0
3 6 3 0
4 12 12 4 0
...
then we obtain the exponential Riordan array [x*exp(x),x], which factors as [x,x]*[exp(x),x] = A132440*A007318. This array is the infinitesimal generator for A116071. A signed version of the array is the infinitesimal generator for A215652. - Peter Bala, Sep 14 2012
a(n,k) = (n-1)!/((n-k)!(k-1)!) if k > n/2 and a(n,k) = (n-1)!/((n-k-1)!k!) otherwise. [Forms 'core' for Pascal's recurrence; gives common term of RHS of T(n,k) = T(n-1,k-1) + T(n-1,k)]. - Jon Perry, Oct 08 2013
Assuming offset 0: T(n, k) = FallingFactorial(n + 1, n) / (k! * (n - k)!). The counterpart using the rising factorial is A356546. - Peter Luschny, Aug 13 2022

Extensions

Edited by N. J. A. Sloane, Oct 07 2007

A011379 a(n) = n^2*(n+1).

Original entry on oeis.org

0, 2, 12, 36, 80, 150, 252, 392, 576, 810, 1100, 1452, 1872, 2366, 2940, 3600, 4352, 5202, 6156, 7220, 8400, 9702, 11132, 12696, 14400, 16250, 18252, 20412, 22736, 25230, 27900, 30752, 33792, 37026, 40460, 44100, 47952, 52022, 56316, 60840
Offset: 0

Views

Author

Glen Burch (gburch(AT)erols.com), Felice Russo

Keywords

Comments

(1) a(n) = sum of second string of n triangular numbers - sum of first n triangular numbers, or the 2n-th partial sum of triangular numbers (A000217) - the n-th partial sum of triangular numbers (A000217). The same for natural numbers gives squares. (2) a(n) = (n-th triangular number)*(the n-th even number) = n(n+1)/2 * (2n). - Amarnath Murthy, Nov 05 2002
Let M(n) be the n X n matrix m(i,j)=1/(i+j+x), let P(n,x) = (Product_{i=0..n-1} i!^2)/det(M(n)). Then P(n,x) is a polynomial with integer coefficients of degree n^2 and a(n) is the coefficient of x^(n^2-1). - Benoit Cloitre, Jan 15 2003
Y values of solutions of the equation: (X-Y)^3-X*Y=0. X values are a(n)=n*(n+1)^2 (see A045991) - Mohamed Bouhamida, May 09 2006
a(2d-1) is the number of self-avoiding walk of length 3 in the d-dimensional hypercubic lattice. - Michael Somos, Sep 06 2006
a(n) mod 10 is periodic 5: repeat [0, 2, 2, 6, 0]. - Mohamed Bouhamida, Sep 05 2009
This sequence is related to A005449 by a(n) = n*A005449(n)-sum(A005449(i), i=0..n-1), and this is the case d=3 in the identity n^2*(d*n+d-2)/2 - Sum_{k=0..n-1} k*(d*k+d-2)/2 = n*(n+d)*(2*d*n+d-3)/6. - Bruno Berselli, Nov 18 2010
Using (n, n+1) to generate a primitive Pythagorean triangle, the sides will be 2*n+1, 2*(n^2+n), and 2*n^2+2*n+1. Inscribing the largest rectangle with integral sides will have sides of length n and n^2+n. Side n is collinear to side 2*n+1 of the triangle and side n^2+n is collinear to side 2*(n^2+n) of the triangle. The areas of theses rectangles are a(n). - J. M. Bergot, Sep 22 2011
a(n+1) is the sum of n-th row of the triangle in A195437. - Reinhard Zumkeller, Nov 23 2011
Partial sums of A049450. - Omar E. Pol, Jan 12 2013
From Jon Perry, May 11 2013: (Start)
Define a 'stable brick triangle' as:
-----
| c |
---------
| a | | b |
----------
with a, b, c > 0 and c <= a + b. This can be visualized as two bricks with a third brick on top. The third brick can only be as strong as a+b, otherwise the wall collapses - for example, (1,2,4) is unstable.
a(n) gives the number of stable brick triangles that can be formed if the two supporting bricks are 1 <= a <= n and 1 <= b <= n: a(n) = Sum_{a=1..n} Sum_{b=1..n} Sum_c 1 = n^3 + n^2 as given in the Adamchuk formula.
So for i=j=n=2 we have 4:
1 2 3 4
2 2 2 2 2 2 2 2
For example, n=2 gives 2 from [a=1,b=1], 3 from both [a=1,b=2] and [a=2,b=1] and 4 from [a=2,b=2] so a(2) = 2 + 3 + 3 + 4 = 12. (End)
Define the infinite square array m(n,k) by m(n,k) = (n-k)^2 if n >= k >= 0 and by m(n,k) = (k+n)*(k-n) if 0 <= n <= k. This contains A120070 below the diagonal. Then a(n) = Sum_{k=0..n} m(n,k) + Sum_{r=0..n} m(r,n), the "hook sum" of the terms to the left of m(n,n) and above m(n,n) with irrelevant (vanishing) terms on the diagonal. - J. M. Bergot, Aug 16 2013
a(n) is the sum of all pairs with repetition drawn from the set of odd numbers 2*n-3. This is similar to A027480 but using the odd integers instead. Example using n=3 gives the odd numbers 1,3,5: 1+1, 1+3, 1+5, 3+3, 3+5,5+5 having a total of 36=a(3). - J. M. Bergot, Apr 05 2016
a(n) is the first Zagreb index of the complete graph K[n+1]. The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternately, it is the sum of the degree sums d(i)+d(j) over all edges ij of the graph. - Emeric Deutsch, Nov 07 2016
a(n-2) is the maximum sigma irregularity over all trees with n vertices. The extremal graphs are stars. (The sigma irregularity of a graph is the sum of squares of the differences between the degrees over all edges of the graph.) - Allan Bickle, Jun 14 2023

Examples

			a(3) = 3^2+3^3 = 36.

References

  • L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, pp. 50, 64.

Links

Crossrefs

Cf. A000217, A000290, A000292, A000330, A000578, A002378, A002411, A002412, A002413, A005449, A013661, A022549, A027480, A045991, A049450, A120070, A126890, A195437, A245334.
Cf. A011379, A181617, A270205 (sigma irregularities of maximal k-degenerate graphs).
Cf. A028724, A052289.

Programs

Formula

a(n) = 2*A002411(n).
a(n) = Sum_{j=1..n} (Sum_{i=1..n} (i+j)), row sums of A126890 skipping numbers in the first column. - Alexander Adamchuk, Oct 12 2004
Sum_{n>0} 1/a(n) = (Pi^2 - 6)/6 = 0.6449340... [Jolley eq 272] - Gary W. Adamson, Dec 22 2006
a(n) = 2*n*binomial(n+1,2) = 2*n*A000217(n). - Arkadiusz Wesolowski, Feb 10 2012
G.f.: 2*x*(1 + 2*x)/(1 - x)^4. - Arkadiusz Wesolowski, Feb 11 2012
a(n) = A000330(n) + A002412(n) = A000292(n) + A002413(n). - Omar E. Pol, Jan 11 2013
a(n) = A245334(n+1,2), n > 0. - Reinhard Zumkeller, Aug 31 2014
Sum_{n>=1} 1/a(n) = A013661-1. - R. J. Mathar, Oct 18 2019 [corrected by Jason Yuen, Aug 04 2024]
Sum_{n>=1} (-1)^(n+1)/a(n) = 1 + Pi^2/12 - 2*log(2). - Amiram Eldar, Jul 04 2020
E.g.f.: exp(x)*x*(2 + 4*x + x^2). - Stefano Spezia, May 20 2021
a(n) = n*A002378(n) = A000578(n) + A000290(n). - J.S. Seneschal, Jun 18 2024

A006002 a(n) = n*(n+1)^2/2.

Original entry on oeis.org

0, 2, 9, 24, 50, 90, 147, 224, 324, 450, 605, 792, 1014, 1274, 1575, 1920, 2312, 2754, 3249, 3800, 4410, 5082, 5819, 6624, 7500, 8450, 9477, 10584, 11774, 13050, 14415, 15872, 17424, 19074, 20825, 22680, 24642, 26714, 28899, 31200, 33620, 36162, 38829, 41624
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

a(n) is the largest number that is not the sum of distinct numbers of form kn+1, k >= 0. - David W. Wilson, Dec 11 1999
Sum of the nontriangular numbers between successive triangular numbers. 1, (2), 3, (4, 5), 6, (7, 8, 9), 10, (11, 12, 13, 14), 15, ... Sum of the terms in brackets. Or sum of n consecutive integers beginning with T(n) + 1, where T(n) = n(n+1)/2. - Amarnath Murthy, Aug 27 2005
Apparently this is also the splittance (as defined by Hammer & Simeone, 1977) of the Kneser graphs of the form K(n+3,2). - Felix Goldberg (felixg(AT)tx.technion.ac.il), Jul 13 2009
Row sums of triangle A159797. - Omar E. Pol, Jul 24 2009
The same results occur when one plots the points (1,3), (3,6), (6,10), (10,15), and so on, for all the triangular numbers and finds the area beneath. Take three consecutive triangular numbers and label them a, b, c; the area created is simply (b-a)*(b+c)/2. Thus for 6,10,15 the area beneath the line defined by the points (6,10) and (10,15) is (10-6)*(10+15)/2 = 50. - J. M. Bergot, Jun 28 2011
Let P = ab where a and b are nonequal prime numbers > 1. Let Q be the product of all divisors of P^n. Q can be expressed as P^k, where k = n*(n+1)^2/2. This follows from the fact that all divisors are of the form a^i*b^j, for i,j from 0 to n. An example is given below. In the more general case, where P is the product of m nonequal prime numbers, k = n*(n+1)^m/2. When m=3, the sequence is the same as A092364. - James A. Raymond & Douglas Raymond, Dec 04 2011
For n > 0: sum of n-th row in A014132, seen as a triangle read by rows. - Reinhard Zumkeller, Dec 12 2012
Partial sums of A005449. - Omar E. Pol, Jan 12 2013
a(n) is the sum of x (or y) coordinates for an n X n square lattice in the upper right quadrant of Z^2 whose corner points are (0, 0), (0, n), (n, 0), and (n, n). - Joseph Wheat, Feb 03 2018
a(n) is the number of permutations of [n+2] that contain exactly 2 elements which are not left-to-right minimal. E.g., the 9 permutations comprising a(2) are 2134, 2143, 3124, 3142, 4123, 4132, 2314, 2413, 3412. - Andy Niedermaier, May 07 2022

Examples

			Let P^n=6^2. The product of the divisors of 36 = 10077796 = 6^9, i.e., for n=2, k=9. - _James A. Raymond_ & Douglas Raymond, Dec 04 2011

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002411: -a(-1-n).
Cf. A000217, A000292, A000330, A014132, A035006, A045991, A092364, A159797, A163274.
Cf. A000914 (partial sums), A005449 (first differences).
Cf. similar sequences of the type n*(n+1)*(n+k)/2 listed in A267370.
Cf. A002378, A027480.
A bisection of A330298.

Programs

Formula

G.f.: x*(x + 2)/(1 - x)^4. - Michael Somos, Jan 30 2004
a(n) = (n + 1) * binomial(n+1, 2). - Zerinvary Lajos, Jan 10 2006
a(n) = A035006(n+1)/4. - Johannes W. Meijer, Feb 04 2010
a(n) = 2*binomial(n+1, 2) + 3*binomial(n+1, 3). - Gary Detlefs, Jun 06 2010
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Harvey P. Dale, Aug 14 2012
a(n) = A000292(n) + A000330(n). - Omar E. Pol, Jan 11 2013
a(n) = A045991(n+1)/2. - J. M. Bergot, Aug 10 2013
a(n) = Sum_{j=1..n} Sum_{i=1..j} (2*j - i + 1). - Wesley Ivan Hurt, Nov 17 2014
a(n) = Sum_{i=0..n} n*(n - i) + i. - Bruno Berselli, Jan 13 2016
a(n) = t(n, A000217(n)), where t(h,k) = A000217(h) + h*k. - Bruno Berselli, Feb 28 2017
Sum_{n>0} 1/a(n) = 4 - Pi^2/3. - Jaume Oliver Lafont, Jul 11 2017 [corrected by Amiram Eldar, Jan 28 2022]
E.g.f.: exp(x)*x*(4 + 5*x + x^2)/2. - Stefano Spezia, May 21 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/6 + 4*log(2) - 4. - Amiram Eldar, Jan 28 2022
From J.S. Seneschal, Jun 27 2024: (Start)
a(n) = (A002378(n)^2/2)/n = (n+1)/2 * A002378(n).
a(n) = A027480(n) - A000217(n). (End)

A050534 Tritriangular numbers: a(n) = binomial(binomial(n,2),2) = n*(n+1)*(n-1)*(n-2)/8.

Original entry on oeis.org

0, 0, 0, 3, 15, 45, 105, 210, 378, 630, 990, 1485, 2145, 3003, 4095, 5460, 7140, 9180, 11628, 14535, 17955, 21945, 26565, 31878, 37950, 44850, 52650, 61425, 71253, 82215, 94395, 107880, 122760, 139128, 157080, 176715, 198135, 221445, 246753, 274170, 303810, 335790
Offset: 0

Views

Author

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 29 1999

Keywords

Comments

"There are n straight lines in a plane, no two of which are parallel and no three of which are concurrent. Their points of intersection being joined, show that the number of new lines drawn is (1/8)n(n-1)(n-2)(n-3)." (Schmall, 1915).
Several different versions of this sequence are possible, beginning with either one, two or three 0's.
If Y is a 3-subset of an n-set X then, for n>=6, a(n-4) is the number of (n-6)-subsets of X which have exactly one element in common with Y. - Milan Janjic, Dec 28 2007
Number of distinct ways to select 2 pairs of objects from a set of n+1 objects, when order doesn't matter. For example, with n = 3 (4 objects), the 3 possibilities are (12)(34), (13)(24), and (14)(23). - Brian Parsonnet, Jan 03 2012
Partial sums of A027480. - J. M. Bergot, Jul 09 2013
For the set {1,2,...,n}, the sum of the 2 smallest elements of all subsets with 3 elements is a(n) (see Bulut et al. link). - Serhat Bulut, Jan 20 2015
a(n) is also the number of subgroups of S_{n+1} (the symmetric group on n+1 elements) that are isomorphic to D_4 (the dihedral group of order 8). - Geoffrey Critzer, Sep 13 2015
a(n) is the coefficient of x1^(n-3)*x2^2 in exponential Bell polynomial B_{n+1}(x1,x2,...) (number of ways to select 2 pairs among n+1 objects, see above), hence its link with A000292 and A001296 (see formula). - Cyril Damamme, Feb 26 2018
Also the number of 4-cycles in the complete graph K_{n+1}. - Eric W. Weisstein, Mar 13 2018
Number of chiral pairs of colorings of the 4 edges or vertices of a square using n or fewer colors. Each member of a chiral pair is a reflection, but not a rotation, of the other. - Robert A. Russell, Oct 20 2020

Examples

			For a(3)=3, the chiral pairs of square colorings are AABC-AACB, ABBC-ACBB, and ABCC-ACCB. - _Robert A. Russell_, Oct 20 2020

References

  • Arthur T. Benjamin and Jennifer Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 154.
  • Louis Comtet, Advanced Combinatorics, Reidel, 1974, Problem 1, page 72.
  • Richard P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.5, case k=2.

Links

Crossrefs

Cf. A000217, A000332, A033487, A107394, A034827, A210569, Second column of triangle A001498.
Cf. similar sequences listed in A241765.
Cf. (square colorings) A006528 (oriented), A002817 (unoriented), A002411 (achiral),
Row 2 of A325006 (orthoplex facets, orthotope vertices) and A337409 (orthotope edges, orthoplex ridges).
Row 4 of A293496 (cycles of n colors using k or fewer colors).

Programs

  • GAP
    List([0..40],n->3*Binomial(n+1,4)); # Muniru A Asiru, Mar 20 2018
  • Magma
    [3*Binomial(n+1, 4): n in [0..40]]; // Vincenzo Librandi, Feb 14 2015
  • Maple
    [seq(binomial(n+1,4)*3,n=0..40)]; # Zerinvary Lajos, Jul 18 2006
  • Mathematica
    Table[Binomial[Binomial[n, 2], 2], {n, 0, 50}] (* Stefan Steinerberger, Apr 08 2006 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 0, 0, 3, 15}, 40] (* Harvey P. Dale, Dec 14 2011 *)
(* Start from Eric W. Weisstein, Mar 13 2018 *)
Binomial[Binomial[Range[0, 20], 2], 2]
Nest[Binomial[#, 2] &, Range[0, 20], 2]
Nest[PolygonalNumber[# - 1] &, Range[0, 20], 2]
CoefficientList[Series[3 x^3/(1 - x)^5, {x, 0, 20}], x]
(* End *)
  • PARI
    a(n)=n*(n+1)*(n-1)*(n-2)/8 \\ Charles R Greathouse IV, Nov 20 2012
  • PARI
    x='x+O('x^100); concat([0, 0, 0], Vec(3*x^3/(1-x)^5)) \\ Altug Alkan, Nov 01 2015
  • Sage
    [(binomial(binomial(n,2),2)) for n in range(0, 39)] # Zerinvary Lajos, Nov 30 2009

Formula

a(n) = 3*binomial(n+1, 4) = 3*A000332(n+1).
From Vladeta Jovovic, May 03 2002: (Start)
Recurrence: a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: 3*x^3 / (1-x)^5. (End)
a(n+1) = T(T(n)) - T(n); a(n+2) = T(T(n)+n) where T is A000217. - Jon Perry, Jun 11 2003
a(n+1) = T(n)^2 - T(T(n)) where T is A000217. - Jon Perry, Jul 23 2003
a(n) = T(T(n-1)-1) where T is A000217. - Jon E. Schoenfield, Dec 14 2014
a(n) = 3*C(n, 4) + 3*C(n, 3), for n>3.
From Alexander Adamchuk, Apr 11 2006: (Start)
a(n) = (1/2)*Sum_{k=1..n} k*(k-1)*(k-2).
a(n) = A033487(n-2)/2, n>1.
a(n) = C(n-1,2)*C(n+1,2)/2, n>2. (End)
a(n) = A052762(n+1)/8. - Zerinvary Lajos, Apr 26 2007
a(n) = (4x^4 - 4x^3 - x^2 + x)/2 where x = floor(n/2)*(-1)^n for n >= 0. - William A. Tedeschi, Aug 24 2010
E.g.f.: x^3*exp(x)*(4+x)/8. - Robert Israel, Nov 01 2015
a(n) = Sum_{k=1..n} Sum_{i=1..k} (n-i-1)*(n-k). - Wesley Ivan Hurt, Sep 12 2017
a(n) = A001296(n-1) - A000292(n-1). - Cyril Damamme, Feb 26 2018
Sum_{n>=3} 1/a(n) = 4/9. - Vaclav Kotesovec, May 01 2018
a(n) = A006528(n) - A002817(n) = (A006528(n) - A002411(n)) / 2 = A002817(n) - A002411(n). - Robert A. Russell, Oct 20 2020
Sum_{n>=3} (-1)^(n+1)/a(n) = 32*log(2)/3 - 64/9. - Amiram Eldar, Jan 09 2022
a(n) = Sum_{k=1..2} (-1)^(k+1)*binomial(n,2-k)*binomial(n,2+k). - Gerry Martens, Oct 09 2022

Extensions

Additional comments from Antreas P. Hatzipolakis, May 03 2002

A046816 Pascal's tetrahedron: entries in 3-dimensional version of Pascal triangle, or trinomial coefficients.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 3, 3, 3, 6, 3, 1, 3, 3, 1, 1, 4, 4, 6, 12, 6, 4, 12, 12, 4, 1, 4, 6, 4, 1, 1, 5, 5, 10, 20, 10, 10, 30, 30, 10, 5, 20, 30, 20, 5, 1, 5, 10, 10, 5, 1, 1, 6, 6, 15, 30, 15, 20, 60, 60, 20, 15, 60, 90, 60, 15, 6, 30, 60, 60, 30, 6, 1, 6, 15, 20, 15, 6, 1
Offset: 0

Views

Author

Lior Manor

Keywords

Comments

Greatest numbers in each 2D triangle form A022916 (multinomial coefficient n!/([n/3]![(n+1)/3]![(n+2)/3]!).) 2D triangle sums are powers of 3. - Gerald McGarvey, Aug 15 2004
T(n,j,k) is the number of lattice paths from (0,0,0) to (n,j,k) with steps (1,0,0), (1,1,0) and (1,1,1). - Dimitri Boscainos, Aug 16 2015
T(n,j,k) is the number of k-dimensional hyperfaces in an n-dimensional hypercube at an edge distance of j from a given vertex. For example, the number of 2D faces in a 3D cube touching a given vertex is T(3,0,2) = 3, and the number of 3D cube 1D edges at a separation of 1 edge from a given vertex is T(3,1,1) = 6. - Eitan Y. Levine, Jul 22 2023
The sums along vertical lines within each slice (when oriented as in the example) give A027907. See "vertical sums" link. - Eitan Y. Levine, May 17 2023
Numbers of ways to classify n circles black, red, or green; classified first by how many circles there are altogether, then by how many are of each color. - J. Lowell, Nov 06 2024

Examples

			The first few slices of the tetrahedron (or pyramid) are:
  1
-----------------
   1
  1 1
-----------------
    1
   2 2
  1 2 1
-----------------
     1 .... Here is the third slice of the pyramid
    3 3
   3 6 3
  1 3 3 1
----------------
...

References

  • Marco Costantini: Metodo per elevare qualsiasi trinomio a qualsiasi potenza. Archimede, rivista per gli insegnanti e i cultori di matematiche pure e applicate, anno XXXVIII ottobre-dicembre 1986, pp. 205-209. [Vincenzo Librandi, Jul 19 2009]

Links

Crossrefs

Cf. A007318, A022916.
Entry [3, 2] in each slice gives A002378, entry [4, 3] in each slice gives A027480, entry [5, 2] in each slice gives A033488, entry [5, 3] in each slice gives A033487.
See A268240 for this read mod 2.
Cf. A013609 (row sums).

Programs

  • Haskell
    a046816 n = a046816_list !! n
a046816_list = concat $ concat $ iterate ([[1],[1,1]] *) [1]
instance Num a => Num [a] where
   fromInteger k = [fromInteger k]
   (p:ps) + (q:qs) = p + q : ps + qs
   ps + qs         = ps ++ qs
   (p:ps) * qs'@(q:qs) = p * q : ps * qs' + [p] * qs
    *                = []
-- Reinhard Zumkeller, Apr 02 2011
  • Maple
    p:= proc(i, j, k) option remember;
      if k<0 or i<0 or i>k or j<0 or j>i then 0
    elif {i, j, k}={0} then 1
    else p(i, j, k-1) +p(i-1, j, k-1) +p(i-1, j-1, k-1)
      fi
    end:
seq(seq(seq(p(i, j, k), j=0..i), i=0..k), k=0..10);
#  Alois P. Heinz, Apr 03 2011
  • Mathematica
    p[i_, j_, k_] := p[i, j, k] = Which[ k<0 || i<0 || i>k || j<0 || j>i, 0, {i, j, k} == {0, 0, 0}, 1, True, p[i, j, k-1] + p[i-1, j, k-1] + p[i-1, j-1, k-1]]; Table[p[i, j, k], {k, 0, 6}, {i, 0, k}, {j, 0, i}] // Flatten (* Jean-François Alcover, Dec 31 2012, translated from Alois P. Heinz's Maple program *)
(* or *)
Flatten[CoefficientList[CoefficientList[CoefficientList[Series[1/(1-x-x*y-x*y*z), {x, 0, 6}], x], y],z]] (* Georg Fischer, May 29 2019 *)
  • Python
    from math import comb as C
def T(n, j, k): return C(n, j) * C(n-j, k)
print([T(n, r-c, c) for n in range(7) for r in range(n+1) for c in range(r+1)]) # Michael S. Branicky, Dec 26 2024

Formula

Coefficients of x, y, z in (x+y+z)^n: Let T'(n; i,j,k) := T(n, j,k) where i = n-(j+k). Then T'(n+1; i,j,k) = T'(n; i-1,j,k)+T'(n; i,j-1,k)+T'(n; i,j,k-1), T'(n; i,j,-1) := 0, T'(n; i,j,k) is invariant under permutations of (i,j,k); T'(0, 0, 0)=1.
T'(n; i,j,k) = n!/(i!*j!*k!) and (x+y+z)^n = Sum_{i+j+k=n; 0 <= i,j,k <= n} T'(n; i,j,k)*x^i*y^j*z^k. Hence Sum_{i+j+k=n; 0 <= i,j,k <= n} T'(n; i,j,k) = 3^n. - Gregory Gerard Wojnar, Oct 08 2020
G.f.: 1/(1-x-x*y-x*y*z). - Georg Fischer, May 29 2019
T(n,j,k) = C(n,j) * C(n-j,k), where C(a,b) are the binomial coefficients, elements of A007318. In particular, T(n,j,0) = C(n,j). - Eitan Y. Levine, Jul 22 2023
(-1)^n * Sum_{i=ceiling(n/k)..n} (-1)^i * T(i*k,n-i,i) = k^n, for n,k > 0. - Eitan Y. Levine, Aug 31 2023
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