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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A000578 The cubes: a(n) = n^3.

Original entry on oeis.org

0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375, 4096, 4913, 5832, 6859, 8000, 9261, 10648, 12167, 13824, 15625, 17576, 19683, 21952, 24389, 27000, 29791, 32768, 35937, 39304, 42875, 46656, 50653, 54872, 59319, 64000, 68921, 74088, 79507
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

a(n) is the sum of the next n odd numbers; i.e., group the odd numbers so that the n-th group contains n elements like this: (1), (3, 5), (7, 9, 11), (13, 15, 17, 19), (21, 23, 25, 27, 29), ...; then each group sum = n^3 = a(n). Also the median of each group = n^2 = mean. As the sum of first n odd numbers is n^2 this gives another proof of the fact that the n-th partial sum = (n(n + 1)/2)^2. - Amarnath Murthy, Sep 14 2002
Total number of triangles resulting from criss-crossing cevians within a triangle so that two of its sides are each n-partitioned. - Lekraj Beedassy, Jun 02 2004. See Propp and Propp-Gubin for a proof.
Also structured triakis tetrahedral numbers (vertex structure 7) (cf. A100175 = alternate vertex); structured tetragonal prism numbers (vertex structure 7) (cf. A100177 = structured prisms); structured hexagonal diamond numbers (vertex structure 7) (cf. A100178 = alternate vertex; A000447 = structured diamonds); and structured trigonal anti-diamond numbers (vertex structure 7) (cf. A100188 = structured anti-diamonds). Cf. A100145 for more on structured polyhedral numbers. - James A. Record (james.record(AT)gmail.com), Nov 07 2004
Schlaefli symbol for this polyhedron: {4, 3}.
Least multiple of n such that every partial sum is a square. - Amarnath Murthy, Sep 09 2005
Draw a regular hexagon. Construct points on each side of the hexagon such that these points divide each side into equally sized segments (i.e., a midpoint on each side or two points on each side placed to divide each side into three equally sized segments or so on), do the same construction for every side of the hexagon so that each side is equally divided in the same way. Connect all such points to each other with lines that are parallel to at least one side of the polygon. The result is a triangular tiling of the hexagon and the creation of a number of smaller regular hexagons. The equation gives the total number of regular hexagons found where n = the number of points drawn + 1. For example, if 1 point is drawn on each side then n = 1 + 1 = 2 and a(n) = 2^3 = 8 so there are 8 regular hexagons in total. If 2 points are drawn on each side then n = 2 + 1 = 3 and a(n) = 3^3 = 27 so there are 27 regular hexagons in total. - Noah Priluck (npriluck(AT)gmail.com), May 02 2007
The solutions of the Diophantine equation: (X/Y)^2 - X*Y = 0 are of the form: (n^3, n) with n >= 1. The solutions of the Diophantine equation: (m^2)*(X/Y)^2k - XY = 0 are of the form: (m*n^(2k + 1), m*n^(2k - 1)) with m >= 1, k >= 1 and n >= 1. The solutions of the Diophantine equation: (m^2)*(X/Y)^(2k + 1) - XY = 0 are of the form: (m*n^(k + 1), m*n^k) with m >= 1, k >= 1 and n >= 1. - Mohamed Bouhamida, Oct 04 2007
Except for the first two terms, the sequence corresponds to the Wiener indices of C_{2n} i.e., the cycle on 2n vertices (n > 1). - K.V.Iyer, Mar 16 2009
Totally multiplicative sequence with a(p) = p^3 for prime p. - Jaroslav Krizek, Nov 01 2009
Sums of rows of the triangle in A176271, n > 0. - Reinhard Zumkeller, Apr 13 2010
One of the 5 Platonic polyhedral (tetrahedral, cube, octahedral, dodecahedral and icosahedral) numbers (cf. A053012). - Daniel Forgues, May 14 2010
Numbers n for which order of torsion subgroup t of the elliptic curve y^2 = x^3 - n is t = 2. - Artur Jasinski, Jun 30 2010
The sequence with the lengths of the Pisano periods mod k is 1, 2, 3, 4, 5, 6, 7, 8, 3, 10, 11, 12, 13, 14, 15, 16, 17, 6, 19, 20, ... for k >= 1, apparently multiplicative and derived from A000027 by dividing every ninth term through 3. Cubic variant of A186646. - R. J. Mathar, Mar 10 2011
The number of atoms in a bcc (body-centered cubic) rhombic hexahedron with n atoms along one edge is n^3 (T. P. Martin, Shells of atoms, eq. (8)). - Brigitte Stepanov, Jul 02 2011
The inverse binomial transform yields the (finite) 0, 1, 6, 6 (third row in A019538 and A131689). - R. J. Mathar, Jan 16 2013
Twice the area of a triangle with vertices at (0, 0), (t(n - 1), t(n)), and (t(n), t(n - 1)), where t = A000217 are triangular numbers. - J. M. Bergot, Jun 25 2013
If n > 0 is not congruent to 5 (mod 6) then A010888(a(n)) divides a(n). - Ivan N. Ianakiev, Oct 16 2013
For n > 2, a(n) = twice the area of a triangle with vertices at points (binomial(n,3),binomial(n+2,3)), (binomial(n+1,3),binomial(n+1,3)), and (binomial(n+2,3),binomial(n,3)). - J. M. Bergot, Jun 14 2014
Determinants of the spiral knots S(4,k,(1,1,-1)). a(k) = det(S(4,k,(1,1,-1))). - Ryan Stees, Dec 14 2014
One of the oldest-known examples of this sequence is shown in the Senkereh tablet, BM 92698, which displays the first 32 terms in cuneiform. - Charles R Greathouse IV, Jan 21 2015
From Bui Quang Tuan, Mar 31 2015: (Start)
We construct a number triangle from the integers 1, 2, 3, ... 2*n-1 as follows. The first column contains all the integers 1, 2, 3, ... 2*n-1. Each succeeding column is the same as the previous column but without the first and last items. The last column contains only n. The sum of all the numbers in the triangle is n^3.
Here is the example for n = 4, where 1 + 2*2 + 3*3 + 4*4 + 3*5 + 2*6 + 7 = 64 = a(4):
1
2 2
3 3 3
4 4 4 4
5 5 5
6 6
7
(End)
For n > 0, a(n) is the number of compositions of n+11 into n parts avoiding parts 2 and 3. - Milan Janjic, Jan 07 2016
Does not satisfy Benford's law [Ross, 2012]. - N. J. A. Sloane, Feb 08 2017
Number of inequivalent face colorings of the cube using at most n colors such that each color appears at least twice. - David Nacin, Feb 22 2017
Consider A = {a,b,c} a set with three distinct members. The number of subsets of A is 8, including {a,b,c} and the empty set. The number of subsets from each of those 8 subsets is 27. If the number of such iterations is n, then the total number of subsets is a(n-1). - Gregory L. Simay, Jul 27 2018
By Fermat's Last Theorem, these are the integers of the form x^k with the least possible value of k such that x^k = y^k + z^k never has a solution in positive integers x, y, z for that k. - Felix Fröhlich, Jul 27 2018

Examples

			For k=3, b(3) = 2 b(2) - b(1) = 4-1 = 3, so det(S(4,3,(1,1,-1))) = 3*3^2 = 27.
For n=3, a(3) = 3 + (3*0^2 + 3*0 + 3*1^2 + 3*1 + 3*2^2 + 3*2) = 27. - _Patrick J. McNab_, Mar 28 2016

References

  • Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See p. 191.
  • John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 43, 64, 81.
  • R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 255; 2nd. ed., p. 269. Worpitzky's identity (6.37).
  • Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §8.6 Figurate Numbers, p. 292.
  • T. Aaron Gulliver, "Sequences from cubes of integers", International Mathematical Journal, 4 (2003), no. 5, 439 - 445. See http://www.m-hikari.com/z2003.html for information about this journal. [I expanded the reference to make this easier to find. - N. J. A. Sloane, Feb 18 2019]
  • J. Propp and A. Propp-Gubin, "Counting Triangles in Triangles", Pi Mu Epsilon Journal (to appear).
  • 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).
  • James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 6-7.
  • D. Wells, You Are A Mathematician, pp. 238-241, Penguin Books 1995.

Links

Crossrefs

(1/12)*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.
For sums of cubes, cf. A000537 (partial sums), A003072, A003325, A024166, A024670, A101102 (fifth partial sums).
Cf. A001158 (inverse Möbius transform), A007412 (complement), A030078(n) (cubes of primes), A048766, A058645 (binomial transform), A065876, A101094, A101097.
Subsequence of A145784.
Cf. A260260 (comment). - Bruno Berselli, Jul 22 2015
Cf. A000292 (tetrahedral numbers), A005900 (octahedral numbers), A006566 (dodecahedral numbers), A006564 (icosahedral numbers).
Cf. A098737 (main diagonal).

Programs

  • Haskell
    a000578 = (^ 3)
a000578_list = 0 : 1 : 8 : zipWith (+)
   (map (+ 6) a000578_list)
   (map (* 3) $ tail $ zipWith (-) (tail a000578_list) a000578_list)
-- Reinhard Zumkeller, Sep 05 2015, May 24 2012, Oct 22 2011
  • Magma
    [ n^3 : n in [0..50] ]; // Wesley Ivan Hurt, Jun 14 2014
  • Magma
    I:=[0,1,8,27]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..45]]; // Vincenzo Librandi, Jul 05 2014
  • Maple
    A000578 := n->n^3;
seq(A000578(n), n=0..50);
isA000578 := proc(r)
    local p;
    if r = 0 or r =1 then
        true;
    else
        for p in ifactors(r)[2] do
            if op(2, p) mod 3 <> 0 then
                return false;
            end if;
        end do:
        true ;
    end if;
end proc: # R. J. Mathar, Oct 08 2013
  • Mathematica
    Table[n^3, {n, 0, 30}] (* Stefan Steinerberger, Apr 01 2006 *)
CoefficientList[Series[x (1 + 4 x + x^2)/(1 - x)^4, {x, 0, 45}], x] (* Vincenzo Librandi, Jul 05 2014 *)
Accumulate[Table[3n^2+3n+1,{n,0,20}]] (* or *) LinearRecurrence[{4,-6,4,-1},{1,8,27,64},20](* Harvey P. Dale, Aug 18 2018 *)
  • Maxima
    A000578(n):=n^3$
makelist(A000578(n),n,0,30); /* Martin Ettl, Nov 03 2012 */
  • PARI
    A000578(n)=n^3 \\ M. F. Hasler, Apr 12 2008
  • PARI
    is(n)=ispower(n,3) \\ Charles R Greathouse IV, Feb 20 2012
  • Python
    A000578_list, m = [], [6, -6, 1, 0]
for _ in range(10**2):
    A000578_list.append(m[-1])
    for i in range(3):
        m[i+1] += m[i] # Chai Wah Wu, Dec 15 2015
  • Scheme
    (define (A000578 n) (* n n n)) ;; Antti Karttunen, Oct 06 2017

Formula

a(n) = Sum_{i=0..n-1} A003215(i).
Multiplicative with a(p^e) = p^(3e). - David W. Wilson, Aug 01 2001
G.f.: x*(1+4*x+x^2)/(1-x)^4. - Simon Plouffe in his 1992 dissertation
Dirichlet generating function: zeta(s-3). - Franklin T. Adams-Watters, Sep 11 2005, Amarnath Murthy, Sep 09 2005
E.g.f.: (1+3*x+x^2)*x*exp(x). - Franklin T. Adams-Watters, Sep 11 2005 - Amarnath Murthy, Sep 09 2005
a(n) = Sum_{i=1..n} (Sum_{j=i..n+i-1} A002024(j,i)). - Reinhard Zumkeller, Jun 24 2007
a(n) = lcm(n, (n - 1)^2) - (n - 1)^2. E.g.: lcm(1, (1 - 1)^2) - (1 - 1)^2 = 0, lcm(2, (2 - 1)^2) - (2 - 1)^2 = 1, lcm(3, (3 - 1)^2) - (3 - 1)^2 = 8, ... - Mats Granvik, Sep 24 2007
Starting (1, 8, 27, 64, 125, ...), = binomial transform of [1, 7, 12, 6, 0, 0, 0, ...]. - Gary W. Adamson, Nov 21 2007
a(n) = A007531(n) + A000567(n). - Reinhard Zumkeller, Sep 18 2009
a(n) = binomial(n+2,3) + 4*binomial(n+1,3) + binomial(n,3). [Worpitzky's identity for cubes. See. e.g., Graham et al., eq. (6.37). - Wolfdieter Lang, Jul 17 2019]
a(n) = n + 6*binomial(n+1,3) = binomial(n,1)+6*binomial(n+1,3). - Ron Knott, Jun 10 2019
A010057(a(n)) = 1. - Reinhard Zumkeller, Oct 22 2011
a(n) = A000537(n) - A000537(n-1), difference between 2 squares of consecutive triangular numbers. - Pierre CAMI, Feb 20 2012
a(n) = A048395(n) - 2*A006002(n). - J. M. Bergot, Nov 25 2012
a(n) = 1 + 7*(n-1) + 6*(n-1)*(n-2) + (n-1)*(n-2)*(n-3). - Antonio Alberto Olivares, Apr 03 2013
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 6. - Ant King Apr 29 2013
a(n) = A000330(n) + Sum_{i=1..n-1} A014105(i), n >= 1. - Ivan N. Ianakiev, Sep 20 2013
a(k) = det(S(4,k,(1,1,-1))) = k*b(k)^2, where b(1)=1, b(2)=2, b(k) = 2*b(k-1) - b(k-2) = b(2)*b(k-1) - b(k-2). - Ryan Stees, Dec 14 2014
For n >= 1, a(n) = A152618(n-1) + A033996(n-1). - Bui Quang Tuan, Apr 01 2015
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Jon Tavasanis, Feb 21 2016
a(n) = n + Sum_{j=0..n-1} Sum_{k=1..2} binomial(3,k)*j^(3-k). - Patrick J. McNab, Mar 28 2016
a(n) = A000292(n-1) * 6 + n. - Zhandos Mambetaliyev, Nov 24 2016
a(n) = n*binomial(n+1, 2) + 2*binomial(n+1, 3) + binomial(n,3). - Tony Foster III, Nov 14 2017
From Amiram Eldar, Jul 02 2020: (Start)
Sum_{n>=1} 1/a(n) = zeta(3) (A002117).
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*zeta(3)/4 (A197070). (End)
From Amiram Eldar, Jan 20 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = cosh(sqrt(3)*Pi/2)/Pi.
Product_{n>=2} (1 - 1/a(n)) = cosh(sqrt(3)*Pi/2)/(3*Pi). (End)
a(n) = Sum_{d|n} sigma_3(d)*mu(n/d) = Sum_{d|n} A001158(d)*A008683(n/d). Moebius transform of sigma_3(n). - Ridouane Oudra, Apr 15 2021

A002411 Pentagonal pyramidal numbers: a(n) = n^2*(n+1)/2.

Original entry on oeis.org

0, 1, 6, 18, 40, 75, 126, 196, 288, 405, 550, 726, 936, 1183, 1470, 1800, 2176, 2601, 3078, 3610, 4200, 4851, 5566, 6348, 7200, 8125, 9126, 10206, 11368, 12615, 13950, 15376, 16896, 18513, 20230, 22050, 23976, 26011, 28158, 30420, 32800, 35301, 37926, 40678
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

a(n) = n^2(n+1)/2 is half the number of colorings of three points on a line with n+1 colors. - R. H. Hardin, Feb 23 2002
Sum of n smallest multiples of n. - Amarnath Murthy, Sep 20 2002
a(n) = number of (n+6)-bit binary sequences with exactly 7 1's none of which is isolated. A 1 is isolated if its immediate neighbor(s) are 0. - David Callan, Jul 15 2004
Also as a(n) = (1/6)*(3*n^3+3*n^2), n > 0: structured trigonal prism numbers (cf. A100177 - structured prisms; A100145 for more on structured numbers). - James A. Record (james.record(AT)gmail.com), Nov 07 2004
Kekulé numbers for certain benzenoids. - Emeric Deutsch, Nov 18 2005
If Y is a 3-subset of an n-set X then, for n >= 5, a(n-4) is the number of 5-subsets of X having at least two elements in common with Y. - Milan Janjic, Nov 23 2007
a(n-1), n >= 2, is the number of ways to have n identical objects in m=2 of altogether n distinguishable boxes (n-2 boxes stay empty). - Wolfdieter Lang, Nov 13 2007
a(n+1) is the convolution of (n+1) and (3n+1). - Paul Barry, Sep 18 2008
The number of 3-character strings from an alphabet of n symbols, if a string and its reversal are considered to be the same.
Partial sums give A001296. - Jonathan Vos Post, Mar 26 2011
a(n-1):=N_1(n), n >= 1, is the number of edges 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
Partial sums of pentagonal numbers A000326. - Reinhard Zumkeller, Jul 07 2012
From Ant King, Oct 23 2012: (Start)
For n > 0, the digital roots of this sequence A010888(A002411(n)) form the purely periodic 9-cycle {1,6,9,4,3,9,7,9,9}.
For n > 0, the units' digits of this sequence A010879(A002411(n)) form the purely periodic 20-cycle {1,6,8,0,5,6,6,8,5,0,6,6,3,0,0,6,1,8,0,0}.
(End)
a(n) is the number of inequivalent ways to color a path graph having 3 nodes using at most n colors. Note, here there is no restriction on the color of adjacent nodes as in the above comment by R. H. Hardin (Feb 23 2002). Also, here the structures are counted up to graph isomorphism, where as in the above comment the "three points on a line" are considered to be embedded in the plane. - Geoffrey Critzer, Mar 20 2013
After 0, row sums of the triangle in A101468. - Bruno Berselli, Feb 10 2014
Latin Square Towers: Take a Latin square of order n, with symbols from 1 to n, and replace each symbol x with a tower of height x. Then the total number of unit cubes used is a(n). - Arun Giridhar, Mar 29 2015
This is the case k = n+4 of b(n,k) = n*((k-2)*n-(k-4))/2, which is the n-th k-gonal number. Therefore, this is the 3rd upper diagonal of the array in A139600. - Luciano Ancora, Apr 11 2015
For n > 0, a(n) is the number of compositions of n+7 into n parts avoiding the part 2. - Milan Janjic, Jan 07 2016
Also the Wiener index of the n-antiprism graph. - Eric W. Weisstein, Sep 07 2017
For n > 0, a(2n+1) is the number of non-isomorphic 5C_m-snakes, where m = 2n+1 or m = 2n (for n >= 2). A kC_n-snake is a connected graph in which the k >= 2 blocks are isomorphic to the cycle C_n and the block-cutpoint graph is a path. - Christian Barrientos, May 15 2019
For n >= 1, a(n-1) is the number of 0°- and 45°-tilted squares that can be drawn by joining points in an n X n lattice. - Paolo Xausa, Apr 13 2021
a(n) is the number of all possible products of n rolls of a six-sided die. This can be easily seen by the recursive formula a(n) = a(n - 1) + 2 * binomial(n, 2) + binomial(n + 1, 2). - Rafal Walczak, Jun 15 2024
a(n) is the number of all triples consisting of nonnegative integers smaller than n such that the sum of the first two integers is less than n. - Ruediger Jehn, Aug 17 2025

Examples

			a(3)=18 because 4 identical balls can be put into m=2 of n=4 distinguishable boxes in binomial(4,2)*(2!/(1!*1!) + 2!/2!) = 6*(2+1) = 18 ways. The m=2 part partitions of 4, namely (1,3) and (2,2), specify the filling of each of the 6 possible two-box choices. - _Wolfdieter Lang_, Nov 13 2007

References

  • V. I. Arnold (ed.), Arnold's Problems, Springer, 2004, comments on Problem 1990-11 (p. 75), pp. 503-510. Numbers N_1.
  • Christian Barrientos, Graceful labelings of cyclic snakes, Ars Combin., Vol. 60 (2001), pp. 85-96.
  • Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 194.
  • S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 166, Table 10.4/I/5).
  • 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. 2.
  • 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).

Links

Crossrefs

Cf. A001296, A011379, A015223, A015224, A014799, A014800, A127739, A132118, A139600.
A006002(n) = -a(-1-n).
a(n) = A093560(n+2, 3), (3, 1)-Pascal column.
A row or column of A132191.
Second column of triangle A103371.
Cf. similar sequences listed in A237616.

Programs

  • GAP
    List([0..45], n->n^2*(n+1)/2); # Muniru A Asiru, Feb 19 2018
  • Haskell
    a002411 n = n * a000217 n  -- Reinhard Zumkeller, Jul 07 2012
  • Magma
    [n^2*(n+1)/2: n in [0..40]]; // Wesley Ivan Hurt, May 25 2014
  • Maple
    seq(n^2*(n+1)/2, n=0..40);
  • Mathematica
    Table[n^2 (n + 1)/2, {n, 0, 40}]
LinearRecurrence[{4, -6, 4, -1}, {0, 1, 6, 18}, 50] (* Harvey P. Dale, Oct 20 2011 *)
Nest[Accumulate, Range[1, 140, 3], 2] (* Vladimir Joseph Stephan Orlovsky, Jan 21 2012 *)
CoefficientList[Series[x (1 + 2 x) / (1 - x)^4, {x, 0, 45}], x] (* Vincenzo Librandi, Jan 08 2016 *)
  • PARI
    a(n)=n^2*(n+1)/2
  • PARI
    concat(0, Vec(x*(1+2*x)/(1-x)^4 + O(x^100))) \\ Altug Alkan, Jan 07 2016

Formula

Average of n^2 and n^3.
G.f.: x*(1+2*x)/(1-x)^4. - Simon Plouffe in his 1992 dissertation
a(n) = n*Sum_{k=0..n} (n-k) = n*Sum_{k=0..n} k. - Paul Barry, Jul 21 2003
a(n) = n*A000217(n). - Xavier Acloque, Oct 27 2003
a(n) = (1/2)*Sum_{j=1..n} Sum_{i=1..n} (i+j) = (1/2)*(n^2+n^3) = (1/2)*A011379(n). - Alexander Adamchuk, Apr 13 2006
Row sums of triangle A127739, triangle A132118; and binomial transform of [1, 5, 7, 3, 0, 0, 0, ...] = (1, 6, 18, 40, 75, ...). - Gary W. Adamson, Aug 10 2007
G.f.: x*F(2,3;1;x). - Paul Barry, Sep 18 2008
Sum_{j>=1} 1/a(j) = hypergeom([1, 1, 1], [2, 3], 1) = -2 + 2*zeta(2) = A195055 - 2. - Stephen Crowley, Jun 28 2009
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=0, a(1)=1, a(2)=6, a(3)=18. - Harvey P. Dale, Oct 20 2011
From Ant King, Oct 23 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 3.
a(n) = (n+1)*(2*A000326(n)+n)/6 = A000292(n) + 2*A000292(n-1).
a(n) = A000330(n)+A000292(n-1) = A000217(n) + 3*A000292(n-1).
a(n) = binomial(n+2,3) + 2*binomial(n+1,3).
(End)
a(n) = (A000330(n) + A002412(n))/2 = (A000292(n) + A002413(n))/2. - Omar E. Pol, Jan 11 2013
a(n) = (24/(n+3)!)*Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*j^(n+3). - Vladimir Kruchinin, Jun 04 2013
Sum_{n>=1} a(n)/n! = (7/2)*exp(1). - Richard R. Forberg, Jul 15 2013
E.g.f.: x*(2 + 4*x + x^2)*exp(x)/2. - Ilya Gutkovskiy, May 31 2016
From R. J. Mathar, Jul 28 2016: (Start)
a(n) = A057145(n+4,n).
a(n) = A080851(3,n-1). (End)
For n >= 1, a(n) = (Sum_{i=1..n} i^2) + Sum_{i=0..n-1} i^2*((i+n) mod 2). - Paolo Xausa, Apr 13 2021
a(n) = Sum_{k=1..n} GCD(k,n) * LCM(k,n). - Vaclav Kotesovec, May 22 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = 2 + Pi^2/6 - 4*log(2). - Amiram Eldar, Jan 03 2022

A005449 Second pentagonal numbers: a(n) = n*(3*n + 1)/2.

Original entry on oeis.org

0, 2, 7, 15, 26, 40, 57, 77, 100, 126, 155, 187, 222, 260, 301, 345, 392, 442, 495, 551, 610, 672, 737, 805, 876, 950, 1027, 1107, 1190, 1276, 1365, 1457, 1552, 1650, 1751, 1855, 1962, 2072, 2185, 2301, 2420, 2542, 2667, 2795, 2926, 3060, 3197, 3337, 3480
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of edges in the join of the complete graph and the cycle graph, both of order n, K_n * C_n. - Roberto E. Martinez II, Jan 07 2002
Also number of cards to build an n-tier house of cards. - Martin Wohlgemuth, Aug 11 2002
The modular form Delta(q) = q*Product_{n>=1} (1-q^n)^24 = q*(1 + Sum_{n>=1} (-1)^n*(q^(n*(3*n-1)/2)+q^(n*(3*n+1)/2)))^24 = q*(1 + Sum_{n>=1} A033999(n)*(q^A000326(n)+q^a(n)))^24. - Jonathan Vos Post, Mar 15 2006
Row sums of triangle A134403.
Bisection of A001318. - Omar E. Pol, Aug 22 2011
Sequence found by reading the line from 0 in the direction 0, 7, ... and the line from 2 in the direction 2, 15, ... in the square spiral whose vertices are the generalized pentagonal numbers, A001318. - Omar E. Pol, Sep 08 2011
A general formula for the n-th second k-gonal number is given by T(n, k) = n*((k-2)*n+k-4)/2, n>=0, k>=5. - Omar E. Pol, Aug 04 2012
Partial sums give A006002. - Denis Borris, Jan 07 2013
A002260 is the following array A read by antidiagonals:
0, 1, 2, 3, 4, 5, ...
0, 1, 2, 3, 4, 5, ...
0, 1, 2, 3, 4, 5, ...
0, 1, 2, 3, 4, 5, ...
0, 1, 2, 3, 4, 5, ...
0, 1, 2, 3, 4, 5, ...
and a(n) is the hook sum: Sum_{k=0..n} A(n,k) + Sum_{r=0..n-1} A(r,n). - R. J. Mathar, Jun 30 2013
From Klaus Purath, May 13 2021: (Start)
This sequence and A000326 provide all integers m such that 24*m + 1 is a square. The union of the two sequences is A001318.
If A is a sequence satisfying the recurrence t(n) = 3*t(n-1) - 2*t(n-2) with the initial values either A(0) = 1, A(1) = n + 2 or A(0) = -1, A(1) = n - 1, then a(n) = (A(i)^2 - A(i-1)*A(i+1))/2^i + n^2 for i>0. (End)
a(n+1) is the number of Dyck paths of size (3,3n+2), i.e., the number of NE lattice paths from (0,0) to (3,3n+2) which stay above the line connecting these points. - Harry Richman, Jul 13 2021
Binomial transform of [0, 2, 3, 0, 0, 0, ...], being a(n) = 2*binomial(n,1) + 3*binomial(n,2). a(3) = 15 = [0, 2, 3, 0] dot [1, 3, 3, 1] = [0 + 6 + 9 + 0]. - Gary W. Adamson, Dec 17 2022
a(n) is the sum of longest side length of all nondegenerate integer-sided triangles with shortest side length n and middle side length (n + 1), n > 0. - Torlach Rush, Feb 04 2024

Examples

			From _Omar E. Pol_, Aug 22 2011: (Start)
Illustration of initial terms:
                                               O
                                             O O
                                 O         O O O
                               O O       O O O O
                     O       O O O     O O O O O
                   O O     O O O O     O O O O O
           O     O O O     O O O O     O O O O O
         O O     O O O     O O O O     O O O O O
    O    O O     O O O     O O O O     O O O O O
    O    O O     O O O     O O O O     O O O O O
    -    ---     -----     -------     ---------
    2     7        15         26           40
(End)

References

  • Henri Cohen, A Course in Computational Algebraic Number Theory, vol. 138 of Graduate Texts in Mathematics, Springer-Verlag, 2000.

Links

Crossrefs

Cf. A016789 (first differences), A006002 (partial sums).
Cf. A000217, A000320, A000326, A001318, A033568, A049451, A101165, A101166.
The generalized pentagonal numbers b*n+3*n*(n-1)/2, for b = 1 through 12, form sequences A000326, this sequence, A045943, A115067, A140090, A140091, A059845, A140672-A140675, A151542.
Cf. numbers of the form n*(n*k-k+4)/2 listed in A226488 (this sequence is the case k=3).
Cf. numbers of the form n*((2*k+1)*n+1)/2 listed in A022289 (this sequence is the case k=1).

Programs

Formula

a(n) = A110449(n, 1) for n>0.
G.f.: x*(2+x)/(1-x)^3. E.g.f.: exp(x)*(2*x + 3*x^2/2). a(n) = n*(3*n + 1)/2. a(-n) = A000326(n). - Michael Somos, Jul 18 2003
a(n) = A001844(n) - A000217(n+1) = A101164(n+2,2) for n>0. - Reinhard Zumkeller, Dec 03 2004
a(n) = Sum_{j=1..n} n+j. - Zerinvary Lajos, Sep 12 2006
a(n) = A126890(n,n). - Reinhard Zumkeller, Dec 30 2006
a(n) = 2*C(3*n,4)/C(3*n,2), n>=1. - Zerinvary Lajos, Jan 02 2007
a(n) = A000217(n) + A000290(n). - Zak Seidov, Apr 06 2008
a(n) = a(n-1) + 3*n - 1 for n>0, a(0)=0. - Vincenzo Librandi, Nov 18 2010
a(n) = A129267(n+5,n). - Philippe Deléham, Dec 21 2011
a(n) = 2*A000217(n) + A000217(n-1). - Philippe Deléham, Mar 25 2013
a(n) = A130518(3*n+1). - Philippe Deléham, Mar 26 2013
a(n) = (12/(n+2)!)*Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*j^(n+2). - Vladimir Kruchinin, Jun 04 2013
a(n) = floor(n/(1-exp(-2/(3*n)))) for n>0. - Richard R. Forberg, Jun 22 2013
a(n) = Sum_{i=1..n} (3*i - 1) for n >= 1. - Wesley Ivan Hurt, Oct 11 2013 [Corrected by Rémi Guillaume, Oct 24 2024]
a(n) = (A000292(6*n+k+1)-A000292(k))/(6*n+1) - A000217(3*n+k+1), for any k >= 0. - Manfred Arens, Apr 26 2015
Sum_{n>=1} 1/a(n) = 6 - Pi/sqrt(3) - 3*log(3) = 0.89036376976145307522... . - Vaclav Kotesovec, Apr 27 2016
a(n) = A000217(2*n) - A000217(n). - Bruno Berselli, Sep 21 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*Pi/sqrt(3) + 4*log(2) - 6. - Amiram Eldar, Jan 18 2021
From Klaus Purath, May 13 2021: (Start)
Partial sums of A016789 for n > 0.
a(n) = 3*n^2 - A000326(n).
a(n) = A000326(n) + n.
a(n) = A002378(n) + A000217(n-1) for n >= 1. [Corrected by Rémi Guillaume, Aug 14 2024] (End)
From Klaus Purath, Jul 14 2021: (Start)
b^2 = 24*a(n) + 1 we get by b^2 = (a(n+1) - a(n-1))^2 = (a(2*n)/n)^2.
a(2*n) = n*(a(n+1) - a(n-1)), n > 0.
a(2*n+1) = n*(a(n+1) - a(n)). (End)
A generalization of Lajos' formula, dated Sep 12 2006, follows. Let SP(k,n) = the n-th second k-gonal number. Then SP(2k+1,n) = Sum_{j=1..n} (k-1)*n+j+k-2. - Charlie Marion, Jul 13 2024
a(n) = Sum_{k = 0..3*n} (-1)^(n+k+1) * binomial(k, 2) * binomial(3*n+k, 2*k). - Peter Bala, Nov 03 2024
For integer m, (6*m + 1)^2*a(n) + a(m) = a((6*m+1)*n + m). - Peter Bala, Jan 09 2025

A045991 a(n) = n^3 - n^2.

Original entry on oeis.org

0, 0, 4, 18, 48, 100, 180, 294, 448, 648, 900, 1210, 1584, 2028, 2548, 3150, 3840, 4624, 5508, 6498, 7600, 8820, 10164, 11638, 13248, 15000, 16900, 18954, 21168, 23548, 26100, 28830, 31744, 34848, 38148, 41650, 45360, 49284, 53428, 57798, 62400, 67240, 72324
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of edges in the line graph of the complete bipartite graph of order 2n, L(K_n,n). - Roberto E. Martinez II, Jan 07 2002
Number of edges of the Cartesian product of two complete graphs K_n X K_n. - Roberto E. Martinez II, Jan 07 2002
That is, number of edges in the n X n rook graph. - Eric W. Weisstein, Jun 20 2017
n such that x^3 + x^2 + n factors over the integers. - James R. Buddenhagen, Apr 19 2005
Also the number of triangles in a 2 X n grid of points and therefore also (n choose 2) * (n choose 1) * 2, or (2n choose 3) - 2*(n choose 3). - Joshua Zucker, Jan 11 2006
Nonnegative X values of solutions to the equation (X-Y)^3-XY=0. To find Y values: b(n)=(n+1)*n^2 (see A011379). I proved that, if(X,Y) is different from (0,0) and m=2, 4, 6, 8, 10, 12,..., then the equation (X-Y)^m-XY=0,... has no solution. - Mohamed Bouhamida, May 10 2006
For n>=1, a(n) is equal to the number of functions f:{1,2,3}->{1,2,...,n} such that for a fixed x in {1,2,3} and a fixed y in {1,2,...,n} we have f(x)<>y. - Aleksandar M. Janjic and Milan Janjic, Mar 13 2007
a(n) equals the coefficient of log(2) in 2F1(n-1,n-1,n+1,-1). - John M. Campbell, Jul 16 2011
Define the infinite square array m(n,k) = (n-k)^2 for 1<=k<=n below the diagonal and m(n,k) = (k+n)(k-n) for 1<=n<=k above the diagonal. Then a(n) = Sum_{k=1..n} m(n,k) + Sum_{r=1..n} m(r,n), the "hook sum" of the terms left from m(n,n) and above m(n,n). - J. M. Bergot, Aug 16 2013
Partial sums of A049451. - Bruno Berselli, Feb 10 2014
Volume of an extruded rectangular brick with side lengths n, n and n-1. - Luciano Ancora, Jun 24 2015

Links

Crossrefs

Cf. A011379, A047929, A114364 (essentially the same).
Cf. A000217, A000290, A000578, A006002, A049451, A146485.
Cf. A002378, A001105, A011379.
Cf. A028724, A052289.

Programs

Formula

G.f.: 2*x^2*(x+2)/(-1+x)^4 = 6/(-1+x)^4+10/(-1+x)^2+14/(-1+x)^3+2/(-1+x). - R. J. Mathar, Nov 19 2007
a(n) = floor(n^5/(n^2+n+1)). - Gary Detlefs, Feb 10 2010
a(n) = 4*binomial(n,2) + 6*binomial(n,3). - Gary Detlefs, Mar 25 2012
a(n+1) = 2*A006002(n). - R. J. Mathar, Oct 31 2012
a(n) = (A000217(n-1)+A000217(n))*(A000217(n-1)-A000217(n-2)). - J. M. Bergot, Oct 31 2012
From Wesley Ivan Hurt, May 19 2015: (Start)
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4).
a(n) = Sum_{k=0..n-1} Sum_{i=n-k-1..n+k-1} i. (End)
Sum_{n>=2} 1/a(n) = 2 - Pi^2/6. - Daniel Suteu, Feb 06 2017
Sum_{n>=2} (-1)^n/a(n) = Pi^2/12 + 2*log(2) - 2. - Amiram Eldar, Jul 05 2020
E.g.f.: exp(x)*x^2*(2 + x). - Stefano Spezia, May 20 2021
Product_{n>=2} (1 - 1/a(n)) = A146485. - Amiram Eldar, Nov 22 2022
From J.S. Seneschal, Jun 21 2024: (Start)
a(n) = (n-1)*A000290(n).
a(n) = n*A002378(n-1).
a(n) = A011379(n) - A001105(n). (End)

A027480 a(n) = n*(n+1)*(n+2)/2.

Original entry on oeis.org

0, 3, 12, 30, 60, 105, 168, 252, 360, 495, 660, 858, 1092, 1365, 1680, 2040, 2448, 2907, 3420, 3990, 4620, 5313, 6072, 6900, 7800, 8775, 9828, 10962, 12180, 13485, 14880, 16368, 17952, 19635, 21420, 23310, 25308, 27417, 29640, 31980, 34440
Offset: 0

Views

Author

Olivier Gérard and Ken Knowlton (kcknowlton(AT)aol.com)

Keywords

Comments

Write the integers in groups: 0; 1,2; 3,4,5; 6,7,8,9; ... and add the groups: a(n) = Sum_{j=0..n} (A000217(n)+j), row sums of the triangular view of A001477. - Asher Auel, Jan 06 2000
With offset = 2, a(n) is the number of edges of the line graph of the complete graph of order n, L(K_n). - Roberto E. Martinez II, Jan 07 2002
Also the total number of pips on a set of dominoes of type n. (A "3" domino set would have 0-0, 0-1, 0-2, 0-3, 1-1, 1-2, 1-3, 2-2, 2-3, 3-3.) - Gerard Schildberger, Jun 26 2003. See A129533 for generalization to n-armed "dominoes". - N. J. A. Sloane, Jan 06 2016
Common sum in an (n+1) X (n+1) magic square with entries (0..n^2-1).
Alternate terms of A057587. - Jeremy Gardiner, Apr 10 2005
If Y is a 3-subset of an n-set X then, for n >= 5, a(n-5) is the number of 4-subsets of X which have exactly one element in common with Y. Also, if Y is a 3-subset of an n-set X then, for n >= 5, a(n-5) is the number of (n-5)-subsets of X which have exactly one element in common with Y. - Milan Janjic, Dec 28 2007
These numbers, starting with 3, are the denominators of the power series f(x) = (1-x)^2 * log(1/(1-x)), if the numerators are kept at 1. This sequence of denominators starts at the term x^3/3. - Miklos Bona, Feb 18 2009
a(n) is the number of triples (w,x,y) having all terms in {0..n} and satisfying at least one of the inequalities x+y < w, y+w < x, w+x < y. - Clark Kimberling, Jun 14 2012
From Martin Licht, Dec 04 2016: (Start)
Let b(n) = (n+1)(n+2)(n+3)/2 (the same sequence, but with a different offset). Then (see Arnold et al., 2006):
b(n) is the dimension of the Nédélec space of the second kind of polynomials of order n over a tetrahedron.
b(n-1) is the dimension of the curl-conforming Nédélec space of the first kind of polynomials of order n with tangential boundary conditions over a tetrahedron.
b(n) is the dimension of the divergence-conforming Nédélec space of the first kind of polynomials of order n with normal boundary conditions over a tetrahedron. (End)
After a(0), the digital root has period 9: repeat [3, 3, 3, 6, 6, 6, 9, 9, 9]. - Peter M. Chema, Jan 19 2017

Examples

			Row sums of n consecutive integers, starting at 0, seen as a triangle:
.
    0 |  0
    3 |  1  2
   12 |  3  4  5
   30 |  6  7  8  9
   60 | 10 11 12 13 14
  105 | 15 16 17 18 19 20

Links

Crossrefs

1/beta(n, 3) in A061928.
Cf. A057587, A006003, A254407.
A row of array in A129533.
Cf. similar sequences of the type n*(n+1)*(n+k)/2 listed in A267370.
Similar sequences are listed in A316224.
Cf. A056923, A281258.
Third column of A003506.
Cf. A000217, A002378, A006002.
A bisection of A330298.

Programs

  • Magma
    [n*(n+1)*(n+2)/2: n in [0..40]]; // Vincenzo Librandi, Nov 14 2014
  • Maple
    [seq(3*binomial(n+2,3),n=0..37)]; # Zerinvary Lajos, Nov 24 2006
a := n -> add((j+n)*(n+2)/3,j=0..n): seq(a(n),n=0..35); # Zerinvary Lajos, Dec 17 2006
  • Mathematica
    Table[(m^3 - m)/2, {m, 36}] (* Zerinvary Lajos, Mar 21 2007 *)
LinearRecurrence[{4,-6,4,-1},{0,3,12,30},40] (* Harvey P. Dale, Oct 10 2012 *)
CoefficientList[Series[3 x / (x - 1)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Nov 14 2014 *)
With[{nn=50},Total/@TakeList[Range[0,(nn(nn+1))/2-1],Range[nn]]] (* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Jun 02 2019 *)
  • PARI
    a(n)=3*binomial(n+2,3) \\ Charles R Greathouse IV, May 23 2011
  • Python
    def a(n): return (n**3+3*n**2+2*n)//2 # _Torlach Rush, Jun 16 2024

Formula

a(n) = a(n-1) + A050534(n) = 3*A000292(n-1) = A050534(n) - A050534(n-1).
a(n) = n*binomial(2+n, 2). - Zerinvary Lajos, Jan 10 2006
a(n) = A007531(n+2)/2. - Zerinvary Lajos, Jul 17 2006
Starting with offset 1 = binomial transform of [3, 9, 9, 3, 0, 0, 0]. - Gary W. Adamson, Oct 25 2007
From R. J. Mathar, Apr 07 2009: (Start)
G.f.: 3*x/(x-1)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)
a(n) = Sum_{i=0..n} n*(n - i) + 2*i. - Bruno Berselli, Jan 13 2016
From Ilya Gutkovskiy, Aug 07 2016: (Start)
E.g.f.: x*(6 + 6*x + x^2)*exp(x)/2.
a(n) = Sum_{k=0..n} A045943(k).
Sum_{n>=1} 1/a(n) = 1/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = (8*log(2) - 5)/2 = 0.2725887222397812... = A016639/10. (End)
a(n-1) = binomial(n^2,2)/n for n > 0. - Jonathan Sondow, Jan 07 2018
For k > 1, Sum_{i=0..n^2-1} (k+i)^2 = (k*n + a(k-1))^2 + A126275(k). - Charlie Marion, Apr 23 2021

A000914 Stirling numbers of the first kind: s(n+2, n).

Original entry on oeis.org

0, 2, 11, 35, 85, 175, 322, 546, 870, 1320, 1925, 2717, 3731, 5005, 6580, 8500, 10812, 13566, 16815, 20615, 25025, 30107, 35926, 42550, 50050, 58500, 67977, 78561, 90335, 103385, 117800, 133672, 151096, 170170, 190995, 213675, 238317, 265031
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Sum of product of unordered pairs of numbers from {1..n+1}.
Number of edges of a complete k-partite graph of order k*(k+1)/2 (A000217), K_1,2,3,...,k. - Roberto E. Martinez II, Oct 18 2001
This sequence holds the x^(n-2) coefficient of the characteristic polynomial of the N X N matrix A formed by MAX(i,j), where i is the row index and j is the column index of element A[i][j], 1 <= i,j <= N. Here N >= 2. - Paul Max Payton, Sep 06 2005
The sequence contains the partial sums of A006002, which represent the areas beneath lines created by the triangular numbers plotted (t(1),t(2)) connected to (t(2),t(3)) then (t(3),t(4))...(t(n-1),t(n)) and the x-axis. - J. M. Bergot, May 05 2012
Number of functions f from [n+2] to [n+2] with f(x)=x for exactly n elements x of [n+2] and f(x)>x for exactly two elements x of [n+2]. To prove this, let the two elements of [n+2] with a larger image be labeled i and j. Note both i and j must be less than n+2. Then there are (n+2-i) choices for f(i) and (n+2-j) choices for f(j). Summing the product of the number of choices over all sets {i,j} gives us "Sum of product of unordered pairs of numbers from {1..n+1}" in the first line of the Comments Section. See the example in the Example Section below. - Dennis P. Walsh, Sep 06 2017
Zhu Shijie gives in his Magnus Opus "Jade Mirror of the Four Unknowns" the problem: "Apples are piled in the form of a triangular pyramid. The top apple is worth 2 and the price of the whole is 1320. Each apple in one layer costs 1 less than an apple in the next layer below." We find the solution 9 to this problem in this sequence 1320 = a(9). Zhu Shijie gave the solution polynomial: "Let the element tian be the number of apples in a side of the base. From the statement we have 31680 for the negative shi, 10 for the positive fang, 21 for the positive first lian, 14 for the positive last lian, and 3 for the positive yu." This translates into the polynomial equation: 3*x^4 + 14*x^3 + 21*x^2 + 10*x - 31680 = 0. - Thomas Scheuerle, Feb 10 2025

Examples

			Examples include E(K_1,2,3) = s(2+2,2) = 11 and E(K_1,2,3,4,5) = s(4+2,4) = 85, where E is the function that counts edges of graphs.
For n=2 the a(2)=11 functions f:[4]->[4] with exactly two f(x)=x and two f(x)>x are given by the 11 image vectors of form <f(1),f(2),f(3),f(4)> that follow: <1,3,4,4>, <1,4,4,4>, <2,2,4,4>, <3,2,4,4>, <4,2,4,4>, <2,3,3,4>, <2,4,3,4>, <3,3,3,4>, <3,4,3,4>, <4,3,3,4>, and <4,4,3,4>. - _Dennis P. Walsh_, Sep 06 2017

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. 833.
  • George E. Andrews, Number Theory, Dover Publications, New York, 1971, p. 4.
  • Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 227, #16.
  • F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226.
  • H. S. Hall and S. R. Knight, Higher Algebra, Fourth Edition, Macmillan, 1891, p. 518.
  • Zhu Shijie, Jade Mirror of the Four Unknowns (Siyuan yujian), Book III Guo Duo Die Gang (Piles of Fruit), Problem number 1, 1303.
  • 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).

Links

Crossrefs

Cf. A000217, A000290, A033428, A033581, A033583, A008275, A052149.
Cf. similar sequences listed in A241765.
Cf. A001296.
Cf. A006325(n+1) (Zhu Shijie's problem number 2 uses a pyramid with square base).

Programs

  • Haskell
    a000914 n = a000914_list !! n
a000914_list = scanl1 (+) a006002_list
-- Reinhard Zumkeller, Mar 25 2014
  • Magma
    [StirlingFirst(n+2, n): n in [0..40]]; // Vincenzo Librandi, May 28 2019
  • Maple
    A000914 := n -> 1/24*(n+1)*n*(n+2)*(3*n+5);
A000914 := proc(n)
    combinat[stirling1](n+2,n) ;
end proc: # R. J. Mathar, May 19 2016
  • Mathematica
    Table[StirlingS1[n+2,n],{n,0,40}] (* Harvey P. Dale, Aug 24 2011 *)
a[ n_] := n (n + 1) (n + 2) (3 n + 5) / 24; (* Michael Somos, Sep 04 2017 *)
  • PARI
    a(n)=sum(i=1,n+1,sum(j=1,n+1,i*j*(i
  • PARI
    a(n)=sum(i=1,n+1,sum(j=1,i-1,i*j)) \\ Charles R Greathouse IV, Apr 07 2015
  • PARI
    a(n) = binomial(n+2, 3)*(3*n+5)/4 \\ Charles R Greathouse IV, Apr 07 2015
  • Sage
    [stirling_number1(n+2, n) for n in range(41)] # Zerinvary Lajos, Mar 14 2009

Formula

a(n) = binomial(n+2, 3)*(3*n+5)/4 = (n+1)*n*(n+2)*(3*n+5)/24.
E.g.f.: exp(x)*x*(48 + 84*x + 32*x^2 + 3*x^3)/24.
G.f.: (2*x+x^2)/(1-x)^5. - Simon Plouffe in his 1992 dissertation.
a(n) = Sum_{i=1..n} i*(i+1)^2/2. - Jon Perry, Jul 31 2003
a(n) = A052149(n+1)/2. - J. M. Bergot, Jun 02 2012
-(3*n+2)*(n-1)*a(n) + (n+2)*(3*n+5)*a(n-1) = 0. - R. J. Mathar, Apr 30 2015
a(n) = a(n-1) + (n+1)*binomial(n+1,2) for n >= 1. - Dennis P. Walsh, Sep 21 2015
a(n) = A001296(-2-n) for all n in Z. - Michael Somos, Sep 04 2017
From Amiram Eldar, Jan 10 2022: (Start)
Sum_{n>=1} 1/a(n) = 162*log(3)/5 - 18*sqrt(3)*Pi/5 - 384/25.
Sum_{n>=1} (-1)^(n+1)/a(n) = 36*sqrt(3)*Pi/5 - 96*log(2)/5 - 636/25. (End)
a(n) = 3*A000332(n+3) - A000292(n). - Yasser Arath Chavez Reyes, Apr 03 2024

Extensions

More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 17 2000
Comments from Michael Somos, Jan 29 2000
Erroneous duplicate of the polynomial formula removed by R. J. Mathar, Sep 15 2009

A014132 Complement of triangular numbers (A000217); also array T(n,k) = ((n+k)^2 + n-k)/2, n, k > 0, read by antidiagonals.

Original entry on oeis.org

2, 4, 5, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 79
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Numbers that are not triangular (nontriangular numbers).
Also definable as follows: a(1)=2; for n>1, a(n) is smallest integer greater than a(n-1) such that the condition "n and a(a(n)) have opposite parities" can always be satisfied. - Benoit Cloitre and Matthew Vandermast, Mar 10 2003
Record values in A256188 that are greater than 1. - Reinhard Zumkeller, Mar 26 2015
From Daniel Forgues, Apr 10 2015: (Start)
With n >= 1, k >= 1:
t(n+k) - k, 1 <= k <= n+k-1, n >= 1;
t(n+k-1) + n, 1 <= n <= n+k-1, k >= 1;
where t(n+k) = t(n+k-1) + (n+k) is the (n+k)-th triangular number, while the number of compositions of n+k into 2 parts is C(n+k-1, 2-1) = n+k-1, the number of nontriangular numbers between t(n+k-1) and t(n+k), just right!
Related to Hilbert's Infinite Hotel:
0) All rooms, numbered through the positive integers, are full;
1) An infinite number of trains, each containing an infinite number of passengers, arrives: i.e., a 2-D lattice of pairs of positive integers;
2) Move occupant of room m, m >= 1, to room t(m) = m*(m+1)/2, where t(m) is the m-th triangular number;
3) Assign n-th passenger from k-th train to room t(n+k-1) + n, 1 <= n <= n+k-1, k >= 1;
4) Everybody has his or her own room, no room is empty, for m >= 1.
If situation 1 happens again, repeat steps 2 and 3, you're back to 4.
(End)
1711 + 2*a(n)*(58 + a(n)) is prime for n<=21. The terms that do not have this property start 29,32,34,43,47,58,59,60,62,63,65,68,70,73,... - Benedict W. J. Irwin, Nov 22 2016
Also numbers k with the property that in the symmetric representation of sigma(k) both Dyck paths have a central peak or both Dyck paths have a central valley. (Cf. A237593.) - Omar E. Pol, Aug 28 2018

Examples

			From _Boris Putievskiy_, Jan 14 2013: (Start)
Start of the sequence as a table (read by antidiagonals, right to left), where the k-th row corresponds to the k-th column of the triangle (shown thereafter):
   2,  4,  7, 11, 16, 22, 29, ...
   5,  8, 12, 17, 23, 30, 38, ...
   9, 13, 18, 24, 31, 39, 48, ...
  14, 19, 25, 32, 40, 49, 59, ...
  20, 26, 33, 41, 50, 60, 71, ...
  27, 34, 42, 51, 61, 72, 84, ...
  35, 43, 52, 62, 73, 85, 98, ...
  (...)
Start of the sequence as a triangle (read by rows), where the i elements of the i-th row are t(i) + 1 up to t(i+1) - 1, i >= 1:
   2;
   4,  5;
   7,  8,  9;
  11, 12, 13, 14;
  16, 17, 18, 19, 20;
  22, 23, 24, 25, 26, 27;
  29, 30, 31, 32, 33, 34, 35;
  (...)
Row number i contains i numbers, where t(i) = i*(i+1)/2:
  t(i) + 1, t(i) + 2, ..., t(i) + i = t(i+1) - 1
(End) [Edited by _Daniel Forgues_, Apr 11 2015]

Links

Crossrefs

Cf. A000217, A006002, A035214, A080036, A002024, A007401, A003057, A114327, A002260, A004736, A118011, A237593.
Cf. A000124 (left edge: quasi-triangular numbers), A000096 (right edge: almost-triangular numbers), A006002 (row sums), A001105 (central terms).
Cf. A242401 (subsequence).
Cf. A004202, A256188.
Cf. A145397 (the non-tetrahedral numbers).

Programs

  • Haskell
    a014132 n = n + round (sqrt $ 2 * fromInteger n)
a014132_list = filter ((== 0) . a010054) [0..]
-- Reinhard Zumkeller, Dec 12 2012
  • Magma
    IsTriangular:=func< n | exists{ k: k in [1..Isqrt(2*n)] | n eq (k*(k+1) div 2)} >; [ n: n in [1..90] | not IsTriangular(n) ]; // Klaus Brockhaus, Jan 04 2011
  • Mathematica
    f[n_] := n + Round[Sqrt[2n]]; Array[f, 71] (* or *)
Complement[ Range[83], Array[ #(# + 1)/2 &, 13]] (* Robert G. Wilson v, Oct 21 2005 *)
DeleteCases[Range[80],?(OddQ[Sqrt[8#+1]]&)] (* _Harvey P. Dale, Jul 24 2021 *)
  • PARI
    a(n)=if(n<1,0,n+(sqrtint(8*n-7)+1)\2)
  • PARI
    isok(n) = !ispolygonal(n,3); \\ Michel Marcus, Mar 01 2016
  • Python
    from math import isqrt
def A014132(n): return n+(isqrt((n<<3)-7)+1>>1) # Chai Wah Wu, Jun 17 2024

Formula

a(n) = n + round(sqrt(2*n)).
a(a(n)) = n + 2*floor(1/2 + sqrt(2n)) + 1.
a(n) = a(n-1) + A035214(n), a(1)=2.
a(n) = A080036(n) - 1.
a(n) = n + A002024(n). - Vincenzo Librandi, Jul 08 2010
A010054(a(n)) = 0. - Reinhard Zumkeller, Dec 10 2012
From Boris Putievskiy, Jan 14 2013: (Start)
a(n) = A007401(n)+1.
a(n) = A003057(n)^2 - A114327(n).
a(n) = ((t+2)^2 + i - j)/2, where
i = n-t*(t+1)/2,
j = (t*t+3*t+4)/2-n,
t = floor((-1+sqrt(8*n-7))/2). (End)
A248952(a(n)) < 0. - Reinhard Zumkeller, Oct 20 2014
a(n) = A256188(A004202(n)). - Reinhard Zumkeller, Mar 26 2015
From Robert Israel, Apr 20 2015 (Start):
a(n) = A118011(n) - n.
G.f.: x/(1-x)^2 + x/(1-x) * Sum(j>=0, x^(j*(j+1)/2)) = x/(1-x)^2 + x^(7/8)/(2-2*x) * Theta2(0,sqrt(x)), where Theta2 is a Jacobi theta function. (End)
G.f. as array: x*y*(2 - 2*y + x^2*y + y^2 - x*(1 + y))/((1 - x)^3*(1 - y)^3). - Stefano Spezia, Apr 22 2024

Extensions

Following Alford Arnold's comment: keyword tabl and correspondent crossrefs added by Reinhard Zumkeller, Dec 12 2012
I restored the original definition. - N. J. A. Sloane, Jan 27 2019

A001181 Number of Baxter permutations of length n (also called Baxter numbers).

Original entry on oeis.org

1, 1, 2, 6, 22, 92, 422, 2074, 10754, 58202, 326240, 1882960, 11140560, 67329992, 414499438, 2593341586, 16458756586, 105791986682, 687782586844, 4517543071924, 29949238543316, 200234184620736, 1349097425104912, 9154276618636016, 62522506583844272
Offset: 0

Views

Author

N. J. A. Sloane, Simon Plouffe

Keywords

Comments

As shown by Dulucq and Guilbert (see for example "Baxter Permutations", Discrete Math., 1998), a(n) is also the number of possible paths for three vicious walkers of length n-1 (a.k.a. "vicious 3-watermelons") [Essam & Guttmann (1995), Eq. (63)] [Jensen (2017), Eqs. (1), (2)]. The proof follows easily by comparing Ollerton's recurrence here and the recurrence in Eq. (60) of Essam & Guttmann. In fact, as discussed by Dulucq and Guilbert, this interpretation of the sequence has been known for a long time. - N. J. A. Sloane, Mar 19 2021; additional references provided by Olivier Gerard, Mar 22 2021.
From Roger Ford, Apr 11 2020: (Start)
a(n) is also the number of meanders with n top arches that as the number of arches is reduced by combining the first and last arches every even number of arches produces a meander.
Example: For n = 4, this meander has this trait.
/\ arches= 8
/\ / \
/ \ --> /\ //\ \
Starting meander: /\ /\ / /\ \ Split and /\/\//\\ ///\\/\\
\ \/ \ \/ / / rotate
\ \ / / bottom arches /\
\ \/ / / \ arches= 7
\ / / /\\ /\
\ / Combine start //\//\\\ //\\/\
\ / first arch with
meander: \/ end of last arch
/\ /\
/\ / \ Combine arches /\ //\\ arches= 6
\ \ / /\ \ again then /\ //\\ ///\\\
\ \ \/ / / rotate and join
\ \ / / <-- /\
\ \/ / //\\ /\ arches= 5
\ / Combine ///\\\ //\\
\/ /\
meander: / \
/ /\ \ Combine /\
\/ \/ <-- //\\ /\/\ arches= 4
/\
Combine /\ //\\ arches= 3
meander: /\ Combine
\/ <-- /\ /\ arches= 2
(End)

Examples

			G.f. = x + 2*x^2 + 6*x^3 + 22*x^4 + 92*x^5 + 422*x^6 + 2074*x^7 + ...
a(4) = 22 since all permutations of length 4 are Baxter except 2413 and 3142. - _Michael Somos_, Jul 19 2002

References

  • Arthur T. Benjamin and Naiomi T. Cameron, Counting on determinants, American Mathematical Monthly, 112.6 (2005): 481-492.
  • W. M. Boyce, Generation of a class of permutations associated with commuting functions, Math. Algorithms, 2 (1967), 19-26.
  • William M. Boyce, Commuting functions with no common fixed point, Transactions of the American Mathematical Society 137 (1969): 77-92.
  • T. Y. Chow, Review of "Bonichon, Nicolas; Bousquet-Mélou, Mireille; Fusy, Éric; Baxter permutations and plane bipolar orientations. Sem. Lothar. Combin. 61A (2009/10), Art. B61Ah, 29 pp.", MathSciNet Review MR2734180 (2011m:05023).
  • S. Dulucq and O. Guibert, Baxter permutations, Proceedings of the 7th Conference on Formal Power Series and Algebraic Combinatorics (Noisy-le-Grand, 1995).
  • J. W. Essam and A. J. Guttmann, Vicious walkers and directed polymer networks in general dimensions, Physical Review E, 52(6), 1995, pp. 5849ff. See (60) and (63).
  • Iwan Jensen, Three friendly walkers, Journal of Physics A: Mathematical and Theoretical, Volume 50:2 (2017), #24003, 14 pages; https://doi.org/10.1088/1751-8121/50/2/024003. See (1), (2).
  • S. Kitaev, Patterns in Permutations and Words, Springer-Verlag, 2011. See p. 399, Table A.7.
  • 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).
  • R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.55.
  • Doron Zeilberger, The method of creative telescoping, J. Symb. Comput. 11.3 (1991): 195-204. See Section 7.8.

Links

Crossrefs

Cf. A001183, A001185, A046996, A006002, A007318, A056939, A359363.

Programs

  • GAP
    Concatenation([1], List([1..30], n-> 2*Sum([1..n], k-> Binomial(n+1,k-1)* Binomial(n+1,k)*Binomial(n+1, k+1) )/(n*(n+1)^2) )); # G. C. Greubel, Jul 24 2019
  • Haskell
    a001181 0 = 1
a001181 n =
   (sum $ map (\k -> product $ map (a007318 (n+1)) [k-1..k+1]) [1..n])
    `div` (a006002 n)
-- Reinhard Zumkeller, Oct 23 2011
  • Magma
    [1] cat [2*(&+[Binomial(n+1,k-1)*Binomial(n+1,k)* Binomial(n+1, k+1): k in [1..n]])/(n*(n+1)^2): n in [1..30]]; // G. C. Greubel, Jul 24 2019
  • Maple
    C := binomial; A001181 := proc(n) local k; add(C(n+1, k-1)*C(n+1, k)* C(n+1, k+1)/ (C(n+1, 1)*C(n+1, 2)), k = 1..n); end;
# second Maple program:
a:= proc(n) option remember; `if`(n<2, 1,
      ((7*n^2+7*n-2)*a(n-1)+8*(n-1)*(n-2)*a(n-2))/((n+2)*(n+3)))
    end:
seq(a(n), n=0..24);  # Alois P. Heinz, Jul 29 2022
  • Mathematica
    A001181[n_] := HypergeometricPFQ[{-1-n, -n, 1-n}, {2, 3}, -1] (* Richard L. Ollerton, Sep 13 2006 *)
a[0]=1; a[1]=1; a[n_] := a[n] = ((7n^2+7n-2)*a[n-1] + 8(n-1)(n-2)*a[n-2]) / ((n+2)(n+3)); Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Oct 28 2015, from 3rd formula *)
  • PARI
    {a(n) = if( n<0, 0, sum( k=1, n, binomial(n+1, k-1) * binomial(n+1, k) * binomial(n+1, k+1) / (binomial(n+1, 1) * binomial(n+1, 2))))}; /* Michael Somos, Jul 19 2002 */
  • Python
    from sympy import binomial as C
def a(n): return sum([(C(n + 1, k - 1)*C(n + 1, k)*C(n + 1, k + 1))/(C(n + 1, 1) * C(n + 1, 2)) for k in range(1, n + 1)]) # Indranil Ghosh, Apr 25 2017
  • Sage
    [1]+[2*sum(binomial(n+1,k-1)*binomial(n+1,k)*binomial(n+1, k+1) for k in (1..n))/(n*(n+1)^2) for n in (1..30)] # G. C. Greubel, Jul 24 2019
print([BaxterPermutations(n).cardinality() for n in range(25)])
# Peter Luschny, May 21 2024

Formula

a(n) = Sum_{k=1..n} C(n+1,k-1) * C(n+1,k) * C(n+1,k+1) / (C(n+1,1) * C(n+1,2)).
(n + 1)*(n + 2)*(n + 3)*(3*n - 2)*a(n) = 2*(n + 1)*(9*n^3 + 3*n^2 - 4*n + 4)*a(n - 1) + (3*n - 1)*(n - 2)*(15*n^2 - 5*n - 14)*a(n - 2) + 8*(3*n + 1)*(n - 2)^2*(n - 3)*a(n - 3), if n>1. [Stanley, 1999] - Michael Somos, Jul 19 2002
From Richard L. Ollerton, Sep 13 2006: (Start)
D-finite with recurrence (n + 2)*(n + 3)*a(n) = (7*n^2 + 7*n - 2)*a(n-1) + 8*(n - 1)*(n - 2)*a(n-2), a(0) = a(1) = 1.
a(n) = hypergeom([-1 - n, -n, 1 - n], [2, 3], -1). (End)
[The coefficients of the polynomials p(n, x) = hypergeom([-1-n, -n, 1-n], [2, 3], -x) are given by the triangular form of A056939. - Peter Luschny, Dec 28 2022]
G.f.: -1 + (1/(3*x^2)) * (x-1 + (1-2*x)*hypergeom([-2/3, 2/3],[1],27*x^2/(1-2*x)^3) - (8*x^3-11*x^2-x)*hypergeom([1/3, 2/3],[2],27*x^2/(1-2*x)^3)/(1-2*x)^2 ). - Mark van Hoeij, Oct 23 2011
a(n) ~ 2^(3*n+5)/(Pi*sqrt(3)*n^4). - Vaclav Kotesovec, Oct 01 2012
0 = +a(n)*(+a(n+1)*(+512*a(n+2) + 2624*a(n+3) - 960*a(n+4)) +a(n+2)*(-1216*a(n+2) + 3368*a(n+3) - 1560*a(n+4)) + a(n+3)*(+600*a(n+3) + 120*a(n+4))) + a(n+1)*(+a(n+1)*(-64*a(n+2) - 1288*a(n+3) + 600*a(n+4)) +a(n+2)*(-136*a(n+2) + 295*a(n+3) - 421*a(n+4)) +a(n+3)*(+161*a(n+3) + 41*a(n+4))) + a(n+2)*(+a(n+2)*(+72*a(n+2) + 17*a(n+3) - 19*a(n+4)) + a(n+3)*(-a(n+3) - a(n+4))) if n>=0. - Michael Somos, Mar 09 2017
G.f.: ((x^3+3*x^2+3*x+1)/(1-8*x)^(3/4)*hypergeom([1/4, 5/4],[2],64*x*(1+x)^3/(8*x-1)^3) - 1 + x)/(3*x^2). - Mark van Hoeij, Nov 05 2023

Extensions

Changed initial term to a(0) = 1 (it was a(0) = 0, but there were compelling reasons to change it). - N. J. A. Sloane, Sep 14 2021

A083374 a(n) = n^2 * (n^2 - 1)/2.

Original entry on oeis.org

0, 6, 36, 120, 300, 630, 1176, 2016, 3240, 4950, 7260, 10296, 14196, 19110, 25200, 32640, 41616, 52326, 64980, 79800, 97020, 116886, 139656, 165600, 195000, 228150, 265356, 306936, 353220, 404550, 461280, 523776, 592416, 667590, 749700, 839160, 936396
Offset: 1

Views

Author

Alan Sutcliffe (alansut(AT)ntlworld.com), Jun 05 2003

Keywords

Comments

Triangular numbers t_n as n runs through the squares.
Partial sums of A055112: If one generated Pythagorean primitive triangles from n, n+1, then the collective areas of n of them would be equal to the numbers in this sequence. The sum of the first three triangles is 6+30+84 = 120 which is the third nonzero term of the sequence. - J. M. Bergot, Jul 14 2011
Second leg of Pythagorean triangles with smallest side a cube: A000578(n)^2 + a(n)^2 = A037270(n)^2. - Martin Renner, Nov 12 2011
a(n) is the number of segments on an n X n grid or geoboard. - Martin Renner, Apr 17 2014
Consider the partitions of 2n into two parts (p,q). Then a(n) is the total volume of the family of rectangular prisms with dimensions p, q and |q-p|. - Wesley Ivan Hurt, Apr 15 2018

References

  • Albert H. Beiler, Recreations in the theory of numbers, New York: Dover, (2nd ed.) 1966, p. 106, table 55.

Links

Crossrefs

Cf. A000217, A000578, A002415, A006002, A006011, A008911, A037270, A047928, A055112.

Programs

Formula

a(n) = (n + 1) * A006002(n).
a(n) = A047928(n)/2 = A002415(n+1)*6 = A006011(n+1)*2 = A008911(n+1)*3. - Zerinvary Lajos, May 09 2007
a(n) = binomial(n^2,2), n>=1. - Zerinvary Lajos, Jan 07 2008
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5) for n>5. - R. J. Mathar, Apr 10 2009
G.f.: -6*x^2*(1+x)/(x-1)^5. - R. J. Mathar, Apr 10 2009
Sum_{n>1} 1/a(n) = (21 - 2*Pi^2)/6. - Enrique Pérez Herrero, Apr 01 2013
a(n) = Sum_{k=0..n-1} k*A000217(2*k+1). - Bruno Berselli, Sep 04 2013
a(n) = 2*A000217(n-1)*A000217(n). - Gionata Neri, May 04 2015
a(n) = Sum_{i=1..n^2-1} i. - Wesley Ivan Hurt, Nov 24 2015
E.g.f.: exp(x)*x^2*(6 + 6*x + x^2)/2. - Stefano Spezia, Jun 06 2021
Sum_{n>=2} (-1)^n/a(n) = Pi^2/6 - 3/2. - Amiram Eldar, Nov 02 2021

Extensions

Corrected and extended by T. D. Noe, Oct 25 2006

A036561 Nicomachus triangle read by rows, T(n, k) = 2^(n - k)*3^k, for 0 <= k <= n.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 8, 12, 18, 27, 16, 24, 36, 54, 81, 32, 48, 72, 108, 162, 243, 64, 96, 144, 216, 324, 486, 729, 128, 192, 288, 432, 648, 972, 1458, 2187, 256, 384, 576, 864, 1296, 1944, 2916, 4374, 6561, 512, 768, 1152, 1728, 2592, 3888, 5832, 8748, 13122, 19683
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

The triangle pertaining to this sequence has the property that every row, every column and every diagonal contains a nontrivial geometric progression. More interestingly every line joining any two elements contains a nontrivial geometric progression. - Amarnath Murthy, Jan 02 2002
Kappraff states (pp. 148-149): "I shall refer to this as Nicomachus' table since an identical table of numbers appeared in the Arithmetic of Nicomachus of Gerasa (circa 150 A.D.)" The table was rediscovered during the Italian Renaissance by Leon Battista Alberti, who incorporated the numbers in dimensions of his buildings and in a system of musical proportions. Kappraff states "Therefore a room could exhibit a 4:6 or 6:9 ratio but not 4:9. This ensured that ratios of these lengths would embody musical ratios". - Gary W. Adamson, Aug 18 2003
After Nichomachus and Alberti several Renaissance authors described this table. See for instance Pierre de la Ramée in 1569 (facsimile of a page of his Arithmetic Treatise in Latin in the links section). - Olivier Gérard, Jul 04 2013
The triangle sums, see A180662 for their definitions, link Nicomachus's table with eleven different sequences, see the crossrefs. It is remarkable that these eleven sequences can be described with simple elegant formulas. The mirror of this triangle is A175840. - Johannes W. Meijer, Sep 22 2010
The diagonal sums Sum_{k} T(n - k, k) give A167762(n + 2). - Michael Somos, May 28 2012
Where d(n) is the divisor count function, then d(T(i,j)) = A003991, the rows of which sum to the tetrahedral numbers A000292(n+1). For example, the sum of the divisors of row 4 of this triangle (i = 4), gives d(16) + d(24) + d(36) + d(54) + d(81) = 5 + 8 + 9 + 8 + 5 = 35 = A000292(5). In fact, where p and q are distinct primes, the aforementioned relationship to the divisor function and tetrahedral numbers can be extended to any triangle of numbers in which the i-th row is of form {p^(i-j)*q^j, 0<=j<=i}; i >= 0 (e.g., A003593, A003595). - Raphie Frank, Nov 18 2012, corrected Dec 07 2012
Sequence (or tree) generated by these rules: 1 is in S, and if x is in S, then 2*x and 3*x are in S, and duplicates are deleted as they occur; see A232559. - Clark Kimberling, Nov 28 2013
Partial sums of rows produce Stirling numbers of the 2nd kind: A000392(n+2) = Sum_{m=1..(n^2+n)/2} a(m). - Fred Daniel Kline, Sep 22 2014
A permutation of A003586. - L. Edson Jeffery, Sep 22 2014
Form a word of length i by choosing a (possibly empty) word on alphabet {0,1} then concatenating a word of length j on alphabet {2,3,4}. T(i,j) is the number of such words. - Geoffrey Critzer, Jun 23 2016
Form of Zorach additive triangle (see A035312) where each number is sum of west and northwest numbers, with the additional condition that each number is GCD of the two numbers immediately below it. - Michel Lagneau, Dec 27 2018

Examples

			The start of the sequence as a triangular array read by rows:
   1
   2   3
   4   6   9
   8  12  18  27
  16  24  36  54  81
  32  48  72 108 162 243
  ...
The start of the sequence as a table T(n,k) n, k > 0:
    1    2    4    8   16   32 ...
    3    6   12   24   48   96 ...
    9   18   36   72  144  288 ...
   27   54  108  216  432  864 ...
   81  162  324  648 1296 2592 ...
  243  486  972 1944 3888 7776 ...
  ...
- _Boris Putievskiy_, Jan 08 2013

References

  • Jay Kappraff, Beyond Measure, World Scientific, 2002, p. 148.
  • Flora R. Levin, The Manual of Harmonics of Nicomachus the Pythagorean, Phanes Press, 1994, p. 114.

Links

Crossrefs

Cf. A001047 (row sums), A000400 (central terms), A013620, A007318.
Cf. A003586, A000079, A000244, A007283, A025197, A005010, A003946, A005051.
Triangle sums (see the comments): A001047 (Row1); A015441 (Row2); A005061 (Kn1, Kn4); A016133 (Kn2, Kn3); A016153 (Fi1, Fi2); A016140 (Ca1, Ca4); A180844 (Ca2, Ca3); A180845 (Gi1, Gi4); A180846 (Gi2, Gi3); A180847 (Ze1, Ze4); A016185 (Ze2, Ze3). - Johannes W. Meijer, Sep 22 2010, Sep 10 2011
Antidiagonal cumulative sum: A000392; square arrays cumulative sum: A160869. Antidiagonal products: 6^A000217; antidiagonal cumulative products: 6^A000292; square arrays products: 6^A005449; square array cumulative products: 6^A006002.

Programs

  • Haskell
    a036561 n k = a036561_tabf !! n !! k
a036561_row n = a036561_tabf !! n
a036561_tabf = iterate (\xs@(x:_) -> x * 2 : map (* 3) xs) [1]
-- Reinhard Zumkeller, Jun 08 2013
  • Magma
    /* As triangle: */ [[(2^(i-j)*3^j)/3: j in [1..i]]: i in [1..10]]; // Vincenzo Librandi, Oct 17 2014
  • Maple
    A036561 := proc(n,k): 2^(n-k)*3^k end:
seq(seq(A036561(n,k),k=0..n),n=0..9);
T := proc(n,k) option remember: if k=0 then 2^n elif k>=1 then procname(n,k-1) + procname(n-1,k-1) fi: end: seq(seq(T(n,k),k=0..n),n=0..9);
# Johannes W. Meijer, Sep 22 2010, Sep 10 2011
  • Mathematica
    Flatten[Table[ 2^(i-j) 3^j, {i, 0, 12}, {j, 0, i} ]] (* Flatten added by Harvey P. Dale, Jun 07 2011 *)
  • PARI
    for(i=0,9,for(j=0,i,print1(3^j<<(i-j)", "))) \\ Charles R Greathouse IV, Dec 22 2011
  • PARI
    {T(n, k) = if( k<0 || k>n, 0, 2^(n - k) * 3^k)} /* Michael Somos, May 28 2012 */

Formula

T(n,k) = A013620(n,k)/A007318(n,k). - Reinhard Zumkeller, May 14 2006
T(n,k) = T(n,k-1) + T(n-1,k-1) for n>=1 and 1<=k<=n with T(n,0) = 2^n for n>=0. - Johannes W. Meijer, Sep 22 2010
T(n,k) = 2^(k-1)*3^(n-1), n, k > 0 read by antidiagonals. - Boris Putievskiy, Jan 08 2013
a(n) = 2^(A004736(n)-1)*3^(A002260(n)-1), n > 0, or a(n) = 2^(j-1)*3^(i-1) n > 0, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor[(-1+sqrt(8*n-7))/2]. - Boris Putievskiy, Jan 08 2013
G.f.: 1/((1-2x)(1-3yx)). - Geoffrey Critzer, Jun 23 2016
T(n,k) = (-1)^n * Sum_{q=0..n} (-1)^q * C(k+3*q, q) * C(n+2*q, n-q). - Marko Riedel, Jul 01 2024
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