cp's OEIS Frontend

Also number of tripods (trees with exactly 3 leaves) on n vertices. - Eric W. Weisstein, Mar 05 2011

Also number of partitions of n+3 into exactly 3 parts; number of partitions of n in which the greatest part is less than or equal to 3; and the number of nonnegative solutions to b + 2c + 3d = n.

Also a(n) gives number of partitions of n+6 into 3 distinct parts and number of partitions of 2n+9 into 3 distinct and odd parts, e.g., 15 = 11 + 3 + 1 = 9 + 5 + 1 = 7 + 5 + 3. - Jon Perry, Jan 07 2004

Also bracelets with n+3 beads 3 of which are red (so there are 2 possibilities with 5 beads).

More generally, the number of partitions of n into at most k parts is also the number of partitions of n+k into k positive parts, the number of partitions of n+k in which the greatest part is k, the number of partitions of n in which the greatest part is less than or equal to k, the number of partitions of n+k(k+1)/2 into exactly k distinct positive parts, the number of nonnegative solutions to b + 2c + 3d + ... + kz = n and the number of nonnegative solutions to 2c + 3d + ... + kz <= n. - Henry Bottomley, Apr 17 2001

Also coefficient of q^n in the expansion of (m choose 3)_q as m goes to infinity. - Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002

From Winston C. Yang (winston(AT)cs.wisc.edu), Apr 30 2002: (Start)

Write 1,2,3,4,... in a hexagonal spiral around 0, then a(n) for n > 0 is formed by the folding points (including the initial 1). The spiral begins:

.

85--84--83--82--81--80

/ \

86 56--55--54--53--52 79

/ / \ \

87 57 33--32--31--30 51 78

/ / / \ \ \

88 58 34 16--15--14 29 50 77

/ / / / \ \ \ \

89 59 35 17 5---4 13 28 49 76

/ / / / / \ \ \ \ \

90 60 36 18 6 0 3 12 27 48 75

/ / / / / / / / / / /

91 61 37 19 7 1---2 11 26 47 74

\ \ \ \ / / / /

62 38 20 8---9--10 25 46 73

\ \ \ / / /

63 39 21--22--23--24 45 72

\ \ / /

64 40--41--42--43--44 71

\ /

65--66--67--68--69--70

.

a(p) is maximal number of hexagons in a polyhex with perimeter at most 2p + 6. (End)

a(n-3) is the number of partitions of n into 3 distinct parts, where 0 is allowed as a part. E.g., at n=9, we can write 8+1+0, 7+2+0, 6+3+0, 4+5+0, 1+2+6, 1+3+5 and 2+3+4, which is a(6)=7. - Jon Perry, Jul 08 2003

a(n) gives number of partitions of n+6 into parts <=3 where each part is used at least once (subtract 6=1+2+3 from n). - Jon Perry, Jul 03 2004

This is also the number of partitions of n+3 into exactly 3 parts (there is a 1-to-1 correspondence between the number of partitions of n+3 in which the greatest part is 3 and the number of partitions of n+3 into exactly three parts). - Graeme McRae, Feb 07 2005

Apply the Riordan array (1/(1-x^3),x) to floor((n+2)/2). - Paul Barry, Apr 16 2005

Also, number of triangles that can be created with odd perimeter 3,5,7,9,11,... with all sides whole numbers. Note that triangles with even perimeter can be generated from the odd ones by increasing each side by 1. E.g., a(1) = 1 because perimeter 3 can make {1,1,1} 1 triangle. a(4) = 3 because perimeter 9 can make {1,4,4} {2,3,4} {3,3,3} 3 possible triangles. - Bruce Love (bruce_love(AT)ofs.edu.sg), Nov 20 2006

Also number of nonnegative solutions of the Diophantine equation x+2*y+3*z=n, cf. Pólya/Szegő reference.

From Vladimir Shevelev, Apr 23 2011: (Start)

Also a(n-3), n >= 3, is the number of non-equivalent necklaces of 3 beads each of them painted by one of n colors.

The sequence {a(n-3), n >= 3} solves the so-called Reis problem about convex k-gons in case k=3 (see our comment to A032279).

a(n-3) (n >= 3) is an essentially unimprovable upper estimate for the number of distinct values of the permanent in (0,1)-circulants of order n with three 1's in every row. (End)

A001399(n) is the number of 3-tuples (w,x,y) having all terms in {0,...,n} and w = 2*x+3*y. - Clark Kimberling, Jun 04 2012

Also, for n >= 3, a(n-3) is the number of the distinct triangles in an n-gon, see the Ngaokrajang links. - Kival Ngaokrajang, Mar 16 2013

Also, a(n) is the total number of 5-curve coin patterns (5C4S type: 5 curves covering full 4 coins and symmetry) packing into fountain of coins base (n+3). See illustration in links. - Kival Ngaokrajang, Oct 16 2013

Also a(n) = half the number of minimal zero sequences for Z_n of length 3 [Ponomarenko]. - N. J. A. Sloane, Feb 25 2014

Also, a(n) equals the number of linearly-independent terms at 2n-th order in the power series expansion of an Octahedral Rotational Energy Surface (cf. Harter & Patterson). - Bradley Klee, Jul 31 2015

Also Molien series for invariants of finite Coxeter groups D_3 and A_3. - N. J. A. Sloane, Jan 10 2016

Number of different distributions of n+6 identical balls in 3 boxes as x,y,z where 0 < x < y < z. - Ece Uslu and Esin Becenen, Jan 11 2016

a(n) is also the number of partitions of 2*n with <= n parts and no part >= 4. The bijection to partitions of n with no part >= 4 is: 1 <-> 2, 2 <-> 1 + 3, 3 <-> 3 + 3 (observing the order of these rules). The <- direction uses the following fact for partitions of 2*n with <= n parts and no part >=4: for each part 1 there is a part 3, and an even number (including 0) of remaining parts 3. - Wolfdieter Lang, May 21 2019

List of the terms in A000567(n>=1), A049450(n>=1), A033428(n>=1), A049451(n>=1), A045944(n>=1), and A003215(n) in nondecreasing order. List of the numbers A056105(n)-1, A056106(n)-1, A056107(n)-1, A056108(n)-1, A056109(n)-1, and A003215(m) with n >= 1 and m >= 0 in nondecreasing order. Numbers of the forms 3n*(n-1)+1, n*(3n-2), n*(3n-1), 3n^2, n*(3n+1), n*(3n+2) with n >= 1 listed in nondecreasing order. Integers m such that lattice points from 1 through m on a hexagonal spiral starting at 1 forms a convex polygon. - Ya-Ping Lu, Jan 24 2024

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

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

From Floor van Lamoen, Jul 21 2001: (Start)

Write 1,2,3,4,... in a hexagonal spiral around 0, then a(n) is the sequence found by reading the line from 0 in the direction 0,2,.... The spiral begins:

.

56--55--54--53--52

/ \

57 33--32--31--30 51

/ / \ \

58 34 16--15--14 29 50

/ / / \ \ \

59 35 17 5---4 13 28 49

/ / / / \ \ \ \

60 36 18 6 0 3 12 27 48

/ / / / / . / / / /

61 37 19 7 1---2 11 26 47

\ \ \ \ . / / /

62 38 20 8---9--10 25 46

\ \ \ . / /

63 39 21--22--23--24 45

\ \ . /

64 40--41--42--43--44

\ .

65--66--67--68--69--70

(End)

Starting with offset 1 = binomial transform of [2, 8, 6, 0, 0, 0, ...]. - Gary W. Adamson, Jan 09 2009

Number of possible pawn moves on an (n+1) X (n+1) chessboard (n=>3). - Johannes W. Meijer, Feb 04 2010

a(n) = A069905(6n-1): Number of partitions of 6*n-1 into 3 parts. - Adi Dani, Jun 04 2011

Even octagonal numbers divided by 4. - Omar E. Pol, Aug 19 2011

Partial sums give A011379. - Omar E. Pol, Jan 12 2013

First differences are A016933; second differences equal 6. - Bob Selcoe, Apr 02 2015

For n >= 1, the continued fraction expansion of sqrt(27*a(n)) is [9n-2; {2, 2n-1, 6, 2n-1, 2, 18n-4}]. - Magus K. Chu, Oct 13 2022

Equals (1, 2, 3, 4, ...) convolved with (1, 2, 1, 2, ...). a(4) = 14 = (1, 2, 3, 4) dot (2, 1, 2, 1) = (2 + 2 + 6 + 4). - Gary W. Adamson, May 01 2009

We observe that is the transform of A032766 by the following transform T: T(u_0,u_1,u_2,u_3,...) = (u_0, u_0+u_1, u_0+u_1+u_2, u_0+u_1+u_2+u_3+u_4,...). In other words, v_p = Sum_{k=0..p} u_k and the g.f. phi_v of is given by phi_v = phi_u/(1-z). - Richard Choulet, Jan 28 2010

Equals row sums of a triangle with (1, 4, 7, 10, ...) in every column, shifted down twice for columns > 1. - Gary W. Adamson, Mar 03 2010

Number of pairs (x,y) with x in {0,...,n}, y odd in {0,...,2n}, and x < y. - Clark Kimberling, Jul 02 2012

Also A049451 and positives A000567 interleaved. - Omar E. Pol, Aug 03 2012

Similar to A001082. Members of this family are A093005, A210977, this sequence, A210978, A181995, A210981, A210982. - Omar E. Pol, Aug 09 2012

Molien series for ternary self-dual codes over GF(3) of length 12n containing 11...1.

(1+x)*(1+x^2) / ((1-x)*(1-x^2)*(1-x^3)) is the Poincaré series [or Poincare series] (or Molien series) for H^*(O_3(q); F_2).

a(n) is the position of the n-th triangular number in the running sum of the (pseudo-Orloj) sequence 1,2,1,2,1,2,1...., cf. A028355. - Wouter Meeussen, Mar 10 2002

a(n) = [a(n-1) + (number of even terms so far in the sequence)]. Example: 14 is [10 + 4 even terms so far in the sequence (they are 0,2,4,10)]. See A096777 for the same construction with odd integers. - Eric Angelini, Aug 05 2007

The number of partitions of 2*n into at most 3 parts. - Colin Barker, Mar 31 2015

Also a(n) equals the number of linearly-independent terms at 2n-th order in the power series expansion of a trigonal Rotational Energy Surface. An optimal basis for the expansion follows either decomposition: g1(x) = (1+x)(1+x^2)g2(x) or g1(x) = (1+x^2)x^(-1)g3(x), where g1(x), g2(x), g3(x) are the generating functions for sequences A007980, A001399, A001840. - Bradley Klee, Aug 06 2015

Also a(n) equals the number of linearly-independent terms at 4n-th order in the power series expansion of the symmetrized weight enumerator of a self-dual code of length n over Z4 that contains a vector (+/-)1^n and has all norms divisible by 8. An optimal basis for the expansion follows the decomposition: g1(x) = (1+x)(1+x^2)g2(x) where g1(x), g2(x) are the generating functions for sequences A007980, A001399. (Cf. Calderbank and Sloane, Corollary 5.) - Bradley Klee, Aug 06 2015

Also, a(n) is equal to the number of partitions of 2n+3 of length 3. Letting n=4, there are a(4)=10 partitions of 2n+3=11 of length 3: (9,1,1), (8,2,1), (7,3,1), (7,2,2), (6,4,1), (6,3,2), (5,5,1), (5,4,2), (5,3,3), (4,4,3). - John M. Campbell, Jan 30 2016

a(n) is the number of partitions of n into parts 1 (of two kinds), part 2 (occurring at most once), and parts 3. - Joerg Arndt, Oct 12 2020

Conjecture: a(n) is the maximum number of pieces a triangle can be cut into by n cevians. - Anton Zakharov, Apr 04 2017

Also, a(n) is the number of graphs which are double-triangle descendants of K_5 with n+6 triangles and 3 more vertices than triangles. See Laradji/Mishna/Yeats reference, proposition 3.6 for details. - Karen A. Yeats, Feb 21 2020

2*a(n) = (2*n)^3 is a perfect cube.

Number of edges of the product of two complete bipartite graphs, each of order 2n, K_n,n x K_n,n - Roberto E. Martinez II, Jan 07 2002

This sequence is related to A049451 by a(n) = n*A049451(n) + sum( A049451(i), i=0..n-1 ) for n>0. - Bruno Berselli, Dec 19 2013

For n>=3, also the detour index of the n-gear graph. - Eric W. Weisstein, Dec 20 2017

For n > 0, this sequence can be obtained by summing consecutive blocks of odd numbers where the n-th block contains the next 2n odd numbers. - Marco Zárate, Jun 15 2025

Interleaving of A049450 and A049451 (for n > 0).

Also, integer values of k*(k+1)/3. - Charles R Greathouse IV, Dec 11 2010

The nonzero coefficients of the expansion of f(a) = Product_{k>=1} (1-a^(2k)), see A194159, occur at the terms of the sequence given above, i.e., f(a) = 1 - a^2 - a^4 + a^10 + a^14 - a^24 - a^30 + a^44 + a^52 - a^70 - a^80 + ... = Sum_{n>=0} (-1)^binomial(n+1,2)*a^A152749(n). - Johannes W. Meijer, Aug 21 2011

Partial sums of A109043. - Reinhard Zumkeller, Mar 31 2012

Nonnegative k such that 12*k+1 is a square. - Vicente Izquierdo Gomez, Jul 22 2013

Equivalently, numbers of the form h*(3*h+1), where h = 0, -1, 1, -2, 2, -3, 3, -4, 4, ... (see also the fifth comment of A062717). - Bruno Berselli, Feb 02 2017

For n > 0, a(n-1) is the sum of the largest parts of the partitions of 2n into two even parts. - Wesley Ivan Hurt, Dec 19 2017

The sequence terms occur as exponents in the expansion of Sum_{n >= 0} q^(n*(n+1)/2) * Product_{k >= n+1} 1 - q^k = 1 - q^2 - q^4 + q^10 + q^14 - q^24 - q^30 + + - - .... - Peter Bala, Dec 15 2024

Sequence terms occur as exponents in the expansions of Sum_{n >= 0} q^(n*(2*n+1)) * Product_{k >= 2*n+2} 1 - q^k = Sum_{n >= 0} q^(n*(2*n-1)) * Product_{k >= 2*n+1} 1 - q^k = 1 - q^2 - q^4 + q^10 + q^14 - q^24 - q^30 + + - - .... - Peter Bala, Jun 23 2025

Partial sums of A152467.

Table starts

.....4......14........30.........52..........80..........114...........154

....14......76.......200........446.........836.........1368..........2134

....26.....280......1184.......3396........7778........15476.........27806

....72....1240......7254......26850.......75792.......178834........372150

...142....4978.....43698.....211374......737460......2076774.......5012084

...362...21626....266812....1689732.....7286542.....24378976......68263780

...778...91058...1639804...13587804....72366820....288237528.....936121220

..1938..393660..10152636..109957140...723237026...3427837106...12911643944

..4366.1686980..63132188..893510530..7258296910..40950125796..178893425570

.10628.7309386.394016210.7288331916.73127883596.491063998130.2488110363432