cp's OEIS Frontend

a(n) is the number of ternary strings of length n that contain at most three 0's.- Enrique Navarrete, Mar 13 2024

For n >= 2, number of palindromic squares of length n whose decimal digits are 0 or 1 and with 9 or fewer 1's.

a(6) > 10^36, if it exists. - Bert Dobbelaere, Jan 18 2020

For some authors, the terms "natural numbers" and "counting numbers" include 0, i.e., refer to the nonnegative integers A001477; the term "whole numbers" frequently also designates the whole set of (signed) integers A001057.

a(n) is smallest positive integer which is consistent with sequence being monotonically increasing and satisfying a(a(n)) = n (cf. A007378).

Inverse Euler transform of A000219.

The rectangular array having A000027 as antidiagonals is the dispersion of the complement of the triangular numbers, A000217 (which triangularly form column 1 of this array). The array is also the transpose of A038722. - Clark Kimberling, Apr 05 2003

For nonzero x, define f(n) = floor(nx) - floor(n/x). Then f=A000027 if and only if x=tau or x=-tau. - Clark Kimberling, Jan 09 2005

Numbers of form (2^i)*k for odd k (i.e., n = A006519(n)*A000265(n)); thus n corresponds uniquely to an ordered pair (i,k) where i=A007814, k=A000265 (with A007814(2n)=A001511(n), A007814(2n+1)=0). - Lekraj Beedassy, Apr 22 2006

If the offset were changed to 0, we would have the following pattern: a(n)=binomial(n,0) + binomial(n,1) for the present sequence (number of regions in 1-space defined by n points), A000124 (number of regions in 2-space defined by n straight lines), A000125 (number of regions in 3-space defined by n planes), A000127 (number of regions in 4-space defined by n hyperplanes), A006261, A008859, A008860, A008861, A008862 and A008863, where the last six sequences are interpreted analogously and in each "... by n ..." clause an offset of 0 has been assumed, resulting in a(0)=1 for all of them, which corresponds to the case of not cutting with a hyperplane at all and therefore having one region. - Peter C. Heinig (algorithms(AT)gmx.de), Oct 19 2006

Define a number of points on a straight line to be in general arrangement when no two points coincide. Then these are the numbers of regions defined by n points in general arrangement on a straight line, when an offset of 0 is assumed. For instance, a(0)=1, since using no point at all leaves one region. The sequence satisfies the recursion a(n) = a(n-1) + 1. This has the following geometrical interpretation: Suppose there are already n-1 points in general arrangement, thus defining the maximal number of regions on a straight line obtainable by n-1 points, and now one more point is added in general arrangement. Then it will coincide with no other point and act as a dividing wall thereby creating one new region in addition to the a(n-1)=(n-1)+1=n regions already there, hence a(n)=a(n-1)+1. Cf. the comments on A000124 for an analogous interpretation. - Peter C. Heinig (algorithms(AT)gmx.de), Oct 19 2006

The sequence a(n)=n (for n=1,2,3) and a(n)=n+1 (for n=4,5,...) gives to the rank (minimal cardinality of a generating set) for the semigroup I_n\S_n, where I_n and S_n denote the symmetric inverse semigroup and symmetric group on [n]. - James East, May 03 2007

The sequence a(n)=n (for n=1,2), a(n)=n+1 (for n=3) and a(n)=n+2 (for n=4,5,...) gives the rank (minimal cardinality of a generating set) for the semigroup PT_n\T_n, where PT_n and T_n denote the partial transformation semigroup and transformation semigroup on [n]. - James East, May 03 2007

"God made the integers; all else is the work of man." This famous quotation is a translation of "Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk," spoken by Leopold Kronecker in a lecture at the Berliner Naturforscher-Versammlung in 1886. Possibly the first publication of the statement is in Heinrich Weber's "Leopold Kronecker," Jahresberichte D.M.V. 2 (1893) 5-31. - Clark Kimberling, Jul 07 2007

Binomial transform of A019590, inverse binomial transform of A001792. - Philippe Deléham, Oct 24 2007

Writing A000027 as N, perhaps the simplest one-to-one correspondence between N X N and N is this: f(m,n) = ((m+n)^2 - m - 3n + 2)/2. Its inverse is given by I(k)=(g,h), where g = k - J(J-1)/2, h = J + 1 - g, J = floor((1 + sqrt(8k - 7))/2). Thus I(1)=(1,1), I(2)=(1,2), I(3)=(2,1) and so on; the mapping I fills the first-quadrant lattice by successive antidiagonals. - Clark Kimberling, Sep 11 2008

a(n) is also the mean of the first n odd integers. - Ian Kent, Dec 23 2008

Equals INVERTi transform of A001906, the even-indexed Fibonacci numbers starting (1, 3, 8, 21, 55, ...). - Gary W. Adamson, Jun 05 2009

These are also the 2-rough numbers: positive integers that have no prime factors less than 2. - Michael B. Porter, Oct 08 2009

Totally multiplicative sequence with a(p) = p for prime p. Totally multiplicative sequence with a(p) = a(p-1) + 1 for prime p. - Jaroslav Krizek, Oct 18 2009

Triangle T(k,j) of natural numbers, read by rows, with T(k,j) = binomial(k,2) + j = (k^2-k)/2 + j where 1 <= j <= k. In other words, a(n) = n = binomial(k,2) + j where k is the largest integer such that binomial(k,2) < n and j = n - binomial(k,2). For example, T(4,1)=7, T(4,2)=8, T(4,3)=9, and T(4,4)=10. Note that T(n,n)=A000217(n), the n-th triangular number. - Dennis P. Walsh, Nov 19 2009

Hofstadter-Conway-like sequence (see A004001): a(n) = a(a(n-1)) + a(n-a(n-1)) with a(1) = 1, a(2) = 2. - Jaroslav Krizek, Dec 11 2009

a(n) is also the dimension of the irreducible representations of the Lie algebra sl(2). - Leonid Bedratyuk, Jan 04 2010

Floyd's triangle read by rows. - Paul Muljadi, Jan 25 2010

Number of numbers between k and 2k where k is an integer. - Giovanni Teofilatto, Mar 26 2010

Generated from a(2n) = r*a(n), a(2n+1) = a(n) + a(n+1), r = 2; in an infinite set, row 2 of the array shown in A178568. - Gary W. Adamson, May 29 2010

1/n = continued fraction [n]. Let barover[n] = [n,n,n,...] = 1/k. Then k - 1/k = n. Example: [2,2,2,...] = (sqrt(2) - 1) = 1/k, with k = (sqrt(2) + 1). Then 2 = k - 1/k. - Gary W. Adamson, Jul 15 2010

Number of n-digit numbers the binary expansion of which contains one run of 1's. - Vladimir Shevelev, Jul 30 2010

From Clark Kimberling, Jan 29 2011: (Start)

Let T denote the "natural number array A000027":

1 2 4 7 ...

3 5 8 12 ...

6 9 13 18 ...

10 14 19 25 ...

T(n,k) = n+(n+k-2)*(n+k-1)/2. See A185787 for a list of sequences based on T, such as rows, columns, diagonals, and sub-arrays. (End)

The Stern polynomial B(n,x) evaluated at x=2. See A125184. - T. D. Noe, Feb 28 2011

The denominator in the Maclaurin series of log(2), which is 1 - 1/2 + 1/3 - 1/4 + .... - Mohammad K. Azarian, Oct 13 2011

As a function of Bernoulli numbers B_n (cf. A027641: (1, -1/2, 1/6, 0, -1/30, 0, 1/42, ...)): let V = a variant of B_n changing the (-1/2) to (1/2). Then triangle A074909 (the beheaded Pascal's triangle) * [1, 1/2, 1/6, 0, -1/30, ...] = the vector [1, 2, 3, 4, 5, ...]. - Gary W. Adamson, Mar 05 2012

Number of partitions of 2n+1 into exactly two parts. - Wesley Ivan Hurt, Jul 15 2013

Integers n dividing u(n) = 2u(n-1) - u(n-2); u(0)=0, u(1)=1 (Lucas sequence A001477). - Thomas M. Bridge, Nov 03 2013

For this sequence, the generalized continued fraction a(1)+a(1)/(a(2)+a(2)/(a(3)+a(3)/(a(4)+...))), evaluates to 1/(e-2) = A194807. - Stanislav Sykora, Jan 20 2014

Engel expansion of e-1 (A091131 = 1.71828...). - Jaroslav Krizek, Jan 23 2014

a(n) is the number of permutations of length n simultaneously avoiding 213, 231 and 321 in the classical sense which are breadth-first search reading words of increasing unary-binary trees. For more details, see the entry for permutations avoiding 231 at A245898. - Manda Riehl, Aug 05 2014

a(n) is also the number of permutations simultaneously avoiding 213, 231 and 321 in the classical sense which can be realized as labels on an increasing strict binary tree with 2n-1 nodes. See A245904 for more information on increasing strict binary trees. - Manda Riehl, Aug 07 2014

a(n) = least k such that 2*Pi - Sum_{h=1..k} 1/(h^2 - h + 3/16) < 1/n. - Clark Kimberling, Sep 28 2014

a(n) = least k such that Pi^2/6 - Sum_{h=1..k} 1/h^2 < 1/n. - Clark Kimberling, Oct 02 2014

Determinants of the spiral knots S(2,k,(1)). a(k) = det(S(2,k,(1))). These knots are also the torus knots T(2,k). - Ryan Stees, Dec 15 2014

As a function, the restriction of the identity map on the nonnegative integers {0,1,2,3...}, A001477, to the positive integers {1,2,3,...}. - M. F. Hasler, Jan 18 2015

See also A131685(k) = smallest positive number m such that c(i) = m (i^1 + 1) (i^2 + 2) ... (i^k+ k) / k! takes integral values for all i>=0: For k=1, A131685(k)=1, which implies that this is a well defined integer sequence. - Alexander R. Povolotsky, Apr 24 2015

a(n) is the number of compositions of n+2 into n parts avoiding the part 2. - Milan Janjic, Jan 07 2016

Does not satisfy Benford's law [Berger-Hill, 2017] - N. J. A. Sloane, Feb 07 2017

Parametrization for the finite multisubsets of the positive integers, where, for p_j the j-th prime, n = Product_{j} p_j^(e_j) corresponds to the multiset containing e_j copies of j ('Heinz encoding' -- see A056239, A003963, A289506, A289507, A289508, A289509). - Christopher J. Smyth, Jul 31 2017

The arithmetic function v_1(n,1) as defined in A289197. - Robert Price, Aug 22 2017

For n >= 3, a(n)=n is the least area that can be obtained for an irregular octagon drawn in a square of n units side, whose sides are parallel to the axes, with 4 vertices that coincide with the 4 vertices of the square, and the 4 remaining vertices having integer coordinates. See Affaire de Logique link. - Michel Marcus, Apr 28 2018

a(n+1) is the order of rowmotion on a poset defined by a disjoint union of chains of length n. - Nick Mayers, Jun 08 2018

Number of 1's in n-th generation of 1-D Cellular Automata using Rules 50, 58, 114, 122, 178, 186, 206, 220, 238, 242, 250 or 252 in the Wolfram numbering scheme, started with a single 1. - Frank Hollstein, Mar 25 2019

(1, 2, 3, 4, 5, ...) is the fourth INVERT transform of (1, -2, 3, -4, 5, ...). - Gary W. Adamson, Jul 15 2019

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

An n X n nonnegative matrix A is primitive (see A070322) iff every element of A^k is > 0 for some power k. If A is primitive then the power which should have all positive entries is <= n^2 - 2n + 2 (Wielandt).

a(n) = Phi_4(n), where Phi_k is the k-th cyclotomic polynomial.

As the positive solution to x=2n+1/x is x=n+sqrt(a(n)), the continued fraction expansion of sqrt(a(n)) is {n; 2n, 2n, 2n, 2n, ...}. - Benoit Cloitre, Dec 07 2001

a(n) is one less than the arithmetic mean of its neighbors: a(n) = (a(n-1) + a(n+1))/2 - 1. E.g., 2 = (1+5)/2 - 1, 5 = (2+10)/2 - 1. - Amarnath Murthy, Jul 29 2003

Equivalently, the continued fraction expansion of sqrt(a(n)) is (n;2n,2n,2n,...). - Franz Vrabec, Jan 23 2006

Number of {12,1*2*,21}-avoiding signed permutations in the hyperoctahedral group.

The number of squares of side 1 which can be drawn without lifting the pencil, starting at one corner of an n X n grid and never visiting an edge twice is n^2-2n+2. - Sébastien Dumortier, Jun 16 2005

Also, numbers m such that m^3 - m^2 is a square, (n*(1 + n^2))^2. - Zak Seidov

1 + 2/2 + 2/5 + 2/10 + ... = Pi*coth Pi [Jolley], see A113319. - Gary W. Adamson, Dec 21 2006

For n >= 1, a(n-1) is the minimal number of choices from an n-set such that at least one particular element has been chosen at least n times or each of the n elements has been chosen at least once. Some games define "matches" this way; e.g., in the classic Parker Brothers, now Hasbro, board game Risk, a(2)=5 is the number of cards of three available types (suits) required to guarantee at least one match of three different types or of three of the same type (ignoring any jokers or wildcards). - Rick L. Shepherd, Nov 18 2007

Positive X values of solutions to the equation X^3 + (X - 1)^2 + X - 2 = Y^2. To prove that X = n^2 + 1: Y^2 = X^3 + (X - 1)^2 + X - 2 = X^3 + X^2 - X - 1 = (X - 1)(X^2 + 2X + 1) = (X - 1)*(X + 1)^2 it means: (X - 1) must be a perfect square, so X = n^2 + 1 and Y = n(n^2 + 2). - Mohamed Bouhamida, Nov 29 2007

{a(k): 0 <= k < 4} = divisors of 10. - Reinhard Zumkeller, Jun 17 2009

Appears in A054413 and A086902 in relation to sequences related to the numerators and denominators of continued fractions convergents to sqrt((2*n)^2/4 + 1), n=1, 2, 3, ... . - Johannes W. Meijer, Jun 12 2010

For n > 0, continued fraction [n,n] = n/a(n); e.g., [5,5] = 5/26. - Gary W. Adamson, Jul 15 2010

The only real solution of the form f(x) = A*x^p with negative p which satisfies f^(m)(x) = f^[-1](x), x >= 0, m >= 1, with f^(m) the m-th derivative and f^[-1] the compositional inverse of f, is obtained for m=2*n, p=p(n)= -(sqrt(a(n))-n) and A=A(n)=(fallfac(p(n),2*n))^(-p(n)/(p(n)+1)), with fallfac(x,k):=Product_{j=0..k-1} (x-j) (falling factorials). See the T. Koshy reference, pp. 263-4 (there are also two solutions for positive p, see the corresponding comment in A087475). - Wolfdieter Lang, Oct 21 2010

n + sqrt(a(n)) = [2*n;2*n,2*n,...] with the regular continued fraction with period 1. This is the even case. For the general case see A087475 with the Schroeder reference and comments. For the odd case see A078370.

a(n-1) counts configurations of non-attacking bishops on a 2 X n strip [Chaiken et al., Ann. Combin. 14 (2010) 419]. - R. J. Mathar, Jun 16 2011

Also numbers k such that 4*k-4 is a square. Hence this sequence is the union of A053755 and A069894. - Arkadiusz Wesolowski, Aug 02 2011

a(n) is also the Moore lower bound on the order, A191595(n), of an (n,5)-cage. - Jason Kimberley, Oct 17 2011

Left edge of the triangle in A195437: a(n+1) = A195437(n,0). - Reinhard Zumkeller, Nov 23 2011

If h (5,17,37,65,101,...) is prime is relatively prime to 6, then h^2-1 is divisible by 24. - Vincenzo Librandi, Apr 14 2014

The identity (4*n^2+2)^2 - (n^2+1)*(4*n)^2 = 4 can be written as A005899(n)^2 - a(n)*A008586(n)^2 = 4. - Vincenzo Librandi, Jun 15 2014

a(n) is also the number of permutations simultaneously avoiding 213 and 321 in the classical sense which can be realized as labels on an increasing strict binary tree with 2n-1 nodes. See A245904 for more information on increasing strict binary trees. - Manda Riehl, Aug 07 2014

a(n-1) is the maximum number of stages in the Gale-Shapley algorithm for finding a stable matching between two sets of n elements given an ordering of preferences for each element (see Gura et al.). - Melvin Peralta, Feb 07 2016

Because of Fermat's little theorem, a(n) is never divisible by 3. - Altug Alkan, Apr 08 2016

For n > 0, if a(n) points are placed inside an n X n square, it will always be the case that at least two of the points will be a distance of sqrt(2) units apart or less. - Melvin Peralta, Jan 21 2017

Also the limit as q->1^- of the unimodal polynomial (1-q^(n*k+1))/(1-q) after making the simplification k=n. The unimodal polynomial is from O'Hara's proof of unimodality of q-binomials after making the restriction to partitions of size <= 1. See G_1(n,k) from arXiv:1711.11252. As the size restriction s increases, G_s->G_infinity=G: the q-binomials. Then substituting k=n and q=1 yields the central binomial coefficients: A000984. - Bryan T. Ek, Apr 11 2018

a(n) is the smallest number congruent to both 1 (mod n) and 2 (mod n+1). - David James Sycamore, Apr 04 2019

a(n) is the number of permutations of 1,2,...,n+1 with exactly one reduced decomposition. - Richard Stanley, Dec 22 2022

From Klaus Purath, Apr 03 2025: (Start)

The odd prime factors of these terms are always of the form 4*k + 1.

All a(n) = D satisfy the Pell equation (k*x)^2 - D*y^2 = -1. The values for k and the solutions x, y can be calculated using the following algorithm: k = n, x(0) = 1, x(1) = 4*D - 1, y(0) = 1, y(1) = 4*D - 3. The two recurrences are of the form (4*D - 2, -1). The solutions x, y of the Pell equations for n = {1 ... 14} are in OEIS.

It follows from the above that this sequence is a subsequence of A031396. (End)

These are Hogben's central polygonal numbers with the (two-dimensional) symbol

2

.P

1 n

The first line cuts the pancake into 2 pieces. For n > 1, the n-th line crosses every earlier line (avoids parallelism) and also avoids every previous line intersection, thus increasing the number of pieces by n. For 16 lines, for example, the number of pieces is 2 + 2 + 3 + 4 + 5 + ... + 16 = 137. These are the triangular numbers plus 1 (cf. A000217).

m = (n-1)(n-2)/2 + 1 is also the smallest number of edges such that all graphs with n nodes and m edges are connected. - Keith Briggs, May 14 2004

Also maximal number of grandchildren of a binary vector of length n+2. E.g., a binary vector of length 6 can produce at most 11 different vectors when 2 bits are deleted.

This is also the order dimension of the (strong) Bruhat order on the finite Coxeter group B_{n+1}. - Nathan Reading (reading(AT)math.umn.edu), Mar 07 2002

Number of 132- and 321-avoiding permutations of {1,2,...,n+1}. - Emeric Deutsch, Mar 14 2002

For n >= 1 a(n) is the number of terms in the expansion of (x+y)*(x^2+y^2)*(x^3+y^3)*...*(x^n+y^n). - Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 28 2003

Also the number of terms in (1)(x+1)(x^2+x+1)...(x^n+...+x+1); see A000140.

Narayana transform (analog of the binomial transform) of vector [1, 1, 0, 0, 0, ...] = A000124; using the infinite lower Narayana triangle of A001263 (as a matrix), N; then N * [1, 1, 0, 0, 0, ...] = A000124. - Gary W. Adamson, Apr 28 2005

Number of interval subsets of {1, 2, 3, ..., n} (cf. A002662). - Jose Luis Arregui (arregui(AT)unizar.es), Jun 27 2006

Define a number of straight lines in the plane to be in general arrangement when (1) no two lines are parallel, (2) there is no point common to three lines. Then these are the maximal numbers of regions defined by n straight lines in general arrangement in the plane. - Peter C. Heinig (algorithms(AT)gmx.de), Oct 19 2006

Note that a(n) = a(n-1) + A000027(n-1). This has the following geometrical interpretation: Suppose there are already n-1 lines in general arrangement, thus defining the maximal number of regions in the plane obtainable by n-1 lines and now one more line is added in general arrangement. Then it will cut each of the n-1 lines and acquire intersection points which are in general arrangement. (See the comments on A000027 for general arrangement with points.) These points on the new line define the maximal number of regions in 1-space definable by n-1 points, hence this is A000027(n-1), where for A000027 an offset of 0 is assumed, that is, A000027(n-1) = (n+1)-1 = n. Each of these regions acts as a dividing wall, thereby creating as many new regions in addition to the a(n-1) regions already there, hence a(n) = a(n-1) + A000027(n-1). Cf. the comments on A000125 for an analogous interpretation. - Peter C. Heinig (algorithms(AT)gmx.de), Oct 19 2006

When constructing a zonohedron, one zone at a time, out of (up to) 3-d non-intersecting parallelepipeds, the n-th element of this sequence is the number of edges in the n-th zone added with the n-th "layer" of parallelepipeds. (Verified up to 10-zone zonohedron, the enneacontahedron.) E.g., adding the 10th zone to the enneacontahedron requires 46 parallel edges (edges in the 10th zone) by looking directly at a 5-valence vertex and counting visible vertices. - Shel Kaphan, Feb 16 2006

Binomial transform of (1, 1, 1, 0, 0, 0, ...) and inverse binomial transform of A072863: (1, 3, 9, 26, 72, 192, ...). - Gary W. Adamson, Oct 15 2007

If Y is a 2-subset of an n-set X then, for n >= 3, a(n-3) is the number of (n-2)-subsets of X which do not have exactly one element in common with Y. - Milan Janjic, Dec 28 2007

Equals row sums of triangle A144328. - Gary W. Adamson, Sep 18 2008

It appears that a(n) is the number of distinct values among the fractions F(i+1)/F(j+1) as j ranges from 1 to n and, for each fixed j, i ranges from 1 to j, where F(i) denotes the i-th Fibonacci number. - John W. Layman, Dec 02 2008

a(n) is the number of subsets of {1,2,...,n} that contain at most two elements. - Geoffrey Critzer, Mar 10 2009

For n >= 2, a(n) gives the number of sets of subsets A_1, A_2, ..., A_n of n = {1, 2, ..., n} such that Meet_{i = 1..n} A_i is empty and Sum_{j in [n]} (|Meet{i = 1..n, i != j} A_i|) is a maximum. - Srikanth K S, Oct 22 2009

The numbers along the left edge of Floyd's triangle. - Paul Muljadi, Jan 25 2010

Let A be the Hessenberg matrix of order n, defined by: A[1,j] = A[i,i]:=1, A[i,i-1] = -1, and A[i,j] = 0 otherwise. Then, for n >= 1, a(n-1) = (-1)^(n-1)*coeff(charpoly(A,x),x). - Milan Janjic, Jan 24 2010

Also the number of deck entries of Euler's ship. See the Meijer-Nepveu link. - Johannes W. Meijer, Jun 21 2010

(1 + x^2 + x^3 + x^4 + x^5 + ...)*(1 + 2x + 3x^2 + 4x^3 + 5x^4 + ...) = (1 + 2x + 4x^2 + 7x^3 + 11x^4 + ...). - Gary W. Adamson, Jul 27 2010

The number of length n binary words that have no 0-digits between any pair of consecutive 1-digits. - Jeffrey Liese, Dec 23 2010

Let b(0) = b(1) = 1; b(n) = max(b(n-1)+n-1, b(n-2)+n-2) then a(n) = b(n+1). - Yalcin Aktar, Jul 28 2011

Also number of triangular numbers so far, for n > 0: a(n) = a(n-1) + Sum(A010054(a(k)): 0 <= k < n), see also A097602, A131073. - Reinhard Zumkeller, Nov 15 2012

Also number of distinct sums of 1 through n where each of those can be + or -. E.g., {1+2,1-2,-1+2,-1-2} = {3,-1,1,-3} and a(2) = 4. - Toby Gottfried, Nov 17 2011

This sequence is complete because the sum of the first n terms is always greater than or equal to a(n+1)-1. Consequently, any nonnegative number can be written as a sum of distinct terms of this sequence. See A204009, A072638. - Frank M Jackson, Jan 09 2012

The sequence is the number of distinct sums of subsets of the nonnegative integers, and its first differences are the positive integers. See A208531 for similar results for the squares. - John W. Layman, Feb 28 2012

Apparently the number of Dyck paths of semilength n+1 in which the sum of the first and second ascents add to n+1. - David Scambler, Apr 22 2013

Without 1 and 2, a(n) equals the terminus of the n-th partial sum of sequence 1, 1, 2. Explanation: 1st partial sums of 1, 1, 2 are 1, 2, 4; 2nd partial sums are 1, 3, 7; 3rd partial sums are 1, 4, 11; 4th partial sums are 1, 5, 16, etc. - Bob Selcoe, Jul 04 2013

Equivalently, numbers of the form 2*m^2+m+1, where m = 0, -1, 1, -2, 2, -3, 3, ... . - Bruno Berselli, Apr 08 2014

For n >= 2: quasi-triangular numbers; the almost-triangular numbers being A000096(n), n >= 2. Note that 2 is simultaneously almost-triangular and quasi-triangular. - Daniel Forgues, Apr 21 2015

n points in general position determine "n choose 2" lines, so A055503(n) <= a(n(n-1)/2). If n > 3, the lines are not in general position and so A055503(n) < a(n(n-1)/2). - Jonathan Sondow, Dec 01 2015

The digital root is period 9 (1, 2, 4, 7, 2, 7, 4, 2, 1), also the digital roots of centered 10-gonal numbers (A062786), for n > 0, A133292. - Peter M. Chema, Sep 15 2016

Partial sums of A028310. - J. Conrad, Oct 31 2016

For n >= 0, a(n) is the number of weakly unimodal sequences of length n over the alphabet {1, 2}. - Armend Shabani, Mar 10 2017

From Eric M. Schmidt, Jul 17 2017: (Start)

Number of sequences (e(1), ..., e(n+1)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) < e(j) != e(k). [Martinez and Savage, 2.4]

Number of sequences (e(1), ..., e(n+1)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) < e(j) and e(i) < e(k). [Martinez and Savage, 2.4]

Number of sequences (e(1), ..., e(n+1)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) >= e(j) != e(k). [Martinez and Savage, 2.4]

(End)

Numbers m such that 8m - 7 is a square. - Bruce J. Nicholson, Jul 24 2017

From Klaus Purath, Jan 29 2020: (Start)

The odd prime factors != 7 occur in an interval of p successive terms either never or exactly twice, while 7 always occurs only once. If a prime factor p appears in a(n) and a(m) within such an interval, then n + m == -1 (mod p). When 7 divides a(n), then 2*n == -1 (mod 7). a(n) is never divisible by the prime numbers given in A003625.

While all prime factors p != 7 can occur to any power, a(n) is never divisible by 7^2. The prime factors are given in A045373. The prime terms of this sequence are given in A055469.

(End)

From Roger Ford, May 10 2021: (Start)

a(n-1) is the greatest sum of arch lengths for the top arches of a semi-meander with n arches. An arch length is the number of arches covered + 1.

/\ The top arch has a length of 3. /\ The top arch has a length of 3.

/ \ Both bottom arches have a //\\ The middle arch has a length of 2.

//\/\\ length of 1. ///\\\ The bottom arch has a length of 1.

Example: for n = 4, a(4-1) = a(3) = 7 /\

//\\

/\ ///\\\ 1 + 3 + 2 + 1 = 7. (End)

a(n+1) is the a(n)-th smallest positive integer that has not yet appeared in the sequence. - Matthew Malone, Aug 26 2021

For n> 0, let the n-dimensional cube {0,1}^n be, provided with the Hamming distance, d. Given an element x in {0,1}^n, a(n) is the number of elements y in {0,1}^n such that d(x, y) <= 2. Example: n = 4. (0,0,0,0), (1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1), (0,0,1,1), (0,1,0,1), (0,1,1,0), (1,0,0,1), (1,0,1,0), (1,1,0,0) are at distance <= 2 from (0,0,0,0), so a(4) = 11. - Yosu Yurramendi, Dec 10 2021

a(n) is the sum of the first three entries of row n of Pascal's triangle. - Daniel T. Martin, Apr 13 2022

a(n-1) is the number of Grassmannian permutations that avoid a pattern, sigma, where sigma is a pattern of size 3 with exactly one descent. For example, sigma is one of the patterns, {132, 213, 231, 312}. - Jessica A. Tomasko, Sep 14 2022

a(n+4) is the number of ways to tile an equilateral triangle of side length 2*n with smaller equilateral triangles of side length n and side length 1. For example, with n=2, there are 22 ways to tile an equilateral triangle of side length 4 with smaller ones of sides 2 and 1, including the one tiling with sixteen triangles of sides 1 and the one tiling with four triangles of sides 2. - Ahmed ElKhatib and Greg Dresden, Aug 19 2024

Define a "hatpin" to be the planar graph consisting of a distinguished point (called the "head") and a semi-infinite line from that point. The maximum number of regions than can be formed by drawing n hatpins is a(n-1). See link for the case n = 4. - N. J. A. Sloane, Jun 25 2025

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