A008615 -id:A008615 - cp's OEIS frontend

A016969 a(n) = 6*n + 5.

5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89, 95, 101, 107, 113, 119, 125, 131, 137, 143, 149, 155, 161, 167, 173, 179, 185, 191, 197, 203, 209, 215, 221, 227, 233, 239, 245, 251, 257, 263, 269, 275, 281, 287, 293, 299, 305, 311, 317, 323, 329, 335

Offset: 0

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0(18).
Exponents e such that x^e + x - 1 is reducible.
First differences of A141631. - Paul Curtz, Sep 12 2008
a(n-1), n >= 1, appears as first column in the triangle A239127 related to the Collatz problem. - Wolfdieter Lang, Mar 14 2014
Odd unlucky numbers in A050505. - Fred Daniel Kline, Feb 25 2017
Intersection of A005408 and A016789. - Bruno Berselli, Apr 26 2018
Numbers that are not divisible by their digital root in base 4. - Amiram Eldar, Nov 24 2022

Muniru A Asiru, Table of n, a(n) for n = 0..3000
Mark W. Coffey, Bernoulli identities, zeta relations, determinant expressions, Mellin transforms, and representation of the Hurwitz numbers, arXiv:1601.01673 [math.NT], 2016.
Tanya Khovanova, Recursive Sequences.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 949.
D. H. Lehmer, Lacunary recurrence formulas for the numbers of Bernoulli and Euler, Annals Math., Vol. 36, No. 3 (1935), pp. 637-649.
Amelia Carolina Sparavigna, The Pentagonal Numbers and their Link to an Integer Sequence which contains the Primes of Form 6n-1, Politecnico di Torino (Italy, 2021).
Amelia Carolina Sparavigna, Binary operations inspired by generalized entropies applied to figurate numbers, Politecnico di Torino (Italy, 2021).
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N)).
William A. Stein, The modular forms database.
Leo Tavares, Illustration: Twin Triangular Frames.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A111863, A007310, A008588, A016921, A016933, A016945, A016957, A049452.
Cf. A050505 (unlucky numbers).
Cf. A000217.

List([0..60],n->6*n+5); # Muniru A Asiru, Nov 24 2018

[ 6*n+5: n in [0..55] ]; // Klaus Brockhaus, Jan 04 2009

6Range[0, 59] + 5 (* or *) NestList[6 + # &, 5, 60] (* Harvey P. Dale, Mar 09 2013 *)

a(n)=6*n+5 \\ Charles R Greathouse IV, Jul 10 2016

(1 to 60).map(6 *  - 1).mkString(", ") // _Alonso del Arte, Nov 23 2018

a(n) = A003415(A003415(A125200(n+1)))/2. - Reinhard Zumkeller, Nov 24 2006
A008615(a(n)) = n+1. - Reinhard Zumkeller, Feb 27 2008
a(n) = A007310(2*n+1); complement of A016921 with respect to A007310. - Reinhard Zumkeller, Oct 02 2008
From Klaus Brockhaus, Jan 04 2009: (Start)
G.f.: (5+x)/(1-x)^2.
a(0) = 5; for n > 0, a(n) = a(n-1)+6. (End)
a(n) = A016921(n)+4 = A016933(n)+3 = A016945(n)+2 = A016957(n)+1. - Klaus Brockhaus, Jan 04 2009
a(n) = floor((12n-1)/2) with offset 1..a(1)=5. - Gary Detlefs, Mar 07 2010
a(n) = 4*(3*n+1) - a(n-1) (with a(0) = 5). - Vincenzo Librandi, Nov 20 2010
a(n) = floor(1/(1/sin(1/n) - n)). - Clark Kimberling, Feb 19 2010
a(n) = 3*Sum_{k = 0..n} binomial(6*n+5, 6*k+2)*Bernoulli(6*k+2). - Michel Marcus, Jan 11 2016
a(n) = A049452(n+1) / (n+1). - Torlach Rush, Nov 23 2018
a(n) = 2*A000217(n+2) - 1 - 2*A000217(n-1). See Twin Triangular Frames illustration. - Leo Tavares, Aug 25 2021
Sum_{n>=0} (-1)^n/a(n) = Pi/6 - sqrt(3)*arccoth(sqrt(3))/3. - Amiram Eldar, Dec 10 2021
E.g.f.: exp(x)*(5 + 6*x). - Stefano Spezia, Feb 14 2025

More terms from Klaus Brockhaus, Jan 04 2009

A016957 a(n) = 6*n + 4.

Original entry on oeis.org

4, 10, 16, 22, 28, 34, 40, 46, 52, 58, 64, 70, 76, 82, 88, 94, 100, 106, 112, 118, 124, 130, 136, 142, 148, 154, 160, 166, 172, 178, 184, 190, 196, 202, 208, 214, 220, 226, 232, 238, 244, 250, 256, 262, 268, 274, 280, 286, 292, 298, 304, 310, 316, 322, 328

Offset: 0

N. J. A. Sloane

Number of 2 X n binary matrices avoiding simultaneously the right-angled numbered polyomino patterns (ranpp) (00;1), (01,1) and (11;0). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1 < i2, j1 < j2 and these elements are in the same relative order as those in the triple (x,y,z). In general, the number of m X n 0-1 matrices in question is given by (n+2)*2^(m-1) + 2*m*(n-1) - 2 for m > 1 and n > 1. - Sergey Kitaev, Nov 12 2004
If Y is a 4-subset of an n-set X then, for n >= 4, a(n-4) is the number of 3-subsets of X having at least two elements in common with Y. - Milan Janjic, Dec 08 2007
4th transversal numbers (or 4-transversal numbers): Numbers of the 4th column of positive numbers in the square array of nonnegative and polygonal numbers A139600. Also, numbers of the 4th column in the square array A057145. - Omar E. Pol, May 02 2008
a(n) is the maximum number such that there exists an edge coloring of the complete graph with a(n) vertices using n colors and every subgraph whose edges are of the same color (subgraph induced by edge color) is planar. - Srikanth K S, Dec 18 2010
Also numbers having two antecedents in the Collatz problem: 12*n+8 and 2*n+1 (respectively A017617(n) and A005408(n)). - Michel Lagneau, Dec 28 2012
a(n) = 6n+4 has three undirected edges e1 = (3n+2, 6n+4), e2 = (6n+4, 12n+8) and e3 = (2n+1, 6n+4) in the Collatz graph of A006370. - Heinz Ebert, Mar 16 2021
Conjecture: this sequence contains some but not all, even numbers with odd abundance A088827. They appear in this sequence at indices A186424(n) - 1. - John Tyler Rascoe, Jul 09 2022

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189. - From N. J. A. Sloane, Dec 01 2012

Heinz Ebert, A Graph Theoretical Approach to the Collatz Problem, arXiv:1905.07575 [math.GM], 2019-2020.
Tanya Khovanova, Recursive Sequences.
Sergey Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory, Vol. 4 (2004), Article A21, 20pp.
Index to sequences related to polygonal numbers.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A000217, A005408, A006370, A008588, A008615, A016921, A016789, A016933, A016945, A016958, A016969, A017617, A017329, A057145, A067412, A088827, A139600, A139606, A157176, A186424.

a016957 = (+ 4) . (* 6)  -- Reinhard Zumkeller, Jul 05 2013

seq(6*n+4, n = 0 .. 50) # Matt C. Anderson, Jun 09 2017

Range[4, 1000, 6] (* Vladimir Joseph Stephan Orlovsky, May 27 2011 *)

makelist(6*n+4, n, 0, 30); /* Martin Ettl, Nov 12 2012 */

a(n)=6*n+4 \\ Charles R Greathouse IV, Jul 10 2016

A008615(a(n)) = n+1. - Reinhard Zumkeller, Feb 27 2008
a(n) = A016789(n)*2. - Omar E. Pol, May 02 2008
A157176(a(n)) = A067412(n+1). - Reinhard Zumkeller, Feb 24 2009
a(n) = sqrt(A016958(n)). - Zerinvary Lajos, Jun 30 2009
a(n) = 2*(6*n+1) - a(n-1) (with a(0)=4). - Vincenzo Librandi, Nov 20 2010
a(n) = floor((sqrt(36*n^2 - 36*n + 1) + 6*n + 1)/2). - Srikanth K S, Dec 18 2010
From Colin Barker, Jan 30 2012: (Start)
G.f.: 2*(2+x)/(1-2*x+x^2).
a(n) = 2*a(n-1) - a(n-2). (End)
A089911(2*a(n)) = 9. - Reinhard Zumkeller, Jul 05 2013
a(n) = 3 * A005408(n) + 1. - Fred Daniel Kline, Oct 24 2015
a(n) = A057145(n+2,4). - R. J. Mathar, Jul 28 2016
a(4*n+2) = 4 * a(n). - Zhandos Mambetaliyev, Sep 22 2018
Sum_{n>=0} (-1)^n/a(n) = sqrt(3)*Pi/18 - log(2)/6. - Amiram Eldar, Dec 10 2021
E.g.f.: 2*exp(x)*(2 + 3*x). - Stefano Spezia, May 29 2024

A016933 a(n) = 6*n + 2.

Original entry on oeis.org

2, 8, 14, 20, 26, 32, 38, 44, 50, 56, 62, 68, 74, 80, 86, 92, 98, 104, 110, 116, 122, 128, 134, 140, 146, 152, 158, 164, 170, 176, 182, 188, 194, 200, 206, 212, 218, 224, 230, 236, 242, 248, 254, 260, 266, 272, 278, 284, 290, 296, 302, 308, 314, 320, 326

Offset: 0

N. J. A. Sloane

Number of 3 X n binary matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01;0), (10;0) and (11;0). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1Sergey Kitaev, Nov 11 2004
Exponents n>1 for which 1 - x + x^n is reducible. - Ron Knott, Oct 13 2016
For the Collatz problem, these are the descenders' values that require division by 2. - Fred Daniel Kline, Jan 19 2017
For n > 3, also the number of (not necessarily maximal) cliques in the n-helm graph. - Eric W. Weisstein, Nov 29 2017

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Gennady Eremin, Infinite matrix of odd natural numbers. A bit about Sophie Germain prime numbers, arXiv:2501.17090 [math.GM], 2025. See p. 9.
Tanya Khovanova, Recursive Sequences.
Sergey Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory, Vol. 4 (2004), Article A21, 20pp.
Sergey Kitaev, On multi-avoidance of right angled numbered polyomino patterns, University of Kentucky Research Reports (2004).
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
Eric Weisstein's World of Mathematics, Clique.
Eric Weisstein's World of Mathematics, Helm Graph.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008588, A016921, A016945, A016957, A016969, A017569, A016777, A016813.

a016933 = (+ 2) . (* 6)  -- Reinhard Zumkeller, Jul 05 2013

a[1]:=2:for n from 2 to 100 do a[n]:=a[n-1]+6 od: seq(a[n], n=1..47); # Zerinvary Lajos, Mar 16 2008

Range[2, 500, 6] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *)
Table[6 n + 2, {n, 0, 20}] (* Eric W. Weisstein, Nov 29 2017 *)
6 Range[0, 20] + 2 (* Eric W. Weisstein, Nov 29 2017 *)
LinearRecurrence[{2, -1}, {8, 14}, {0, 20}] (* Eric W. Weisstein, Nov 29 2017 *)
CoefficientList[Series[2 (1 + 2 x)/(-1 + x)^2, {x, 0, 20}], x] (* Eric W. Weisstein, Nov 29 2017 *)

a(n)=6*n+2 \\ Charles R Greathouse IV, Jul 10 2016

[i+2 for i in range(280) if gcd(i,6) == 6] # Zerinvary Lajos, May 20 2009

A008615(a(n)) = n+1. - Reinhard Zumkeller, Feb 27 2008
A157176(a(n)) = A013730(n). - Reinhard Zumkeller, Feb 24 2009
A089911(2*a(n)) = 3. - Reinhard Zumkeller, Jul 05 2013
a(n) = 2*(6*n-1) - a(n-1) (with a(0)=2). - Vincenzo Librandi, Nov 20 2010
G.f.: 2*(1+2*x)/(1-x)^2. - Colin Barker, Jan 08 2012
a(n) = (3 * A016813(n) + 1) / 2.- Fred Daniel Kline, Jan 20 2017
a(n) = A016789(A005843(n)). - Felix Fröhlich, Jan 20 2017
Sum_{n>=0} (-1)^n/a(n) = sqrt(3)*Pi/18 + log(2)/6. - Amiram Eldar, Dec 10 2021
a(n) = 2 * A016777(n). - Alois P. Heinz, Dec 27 2023
From Elmo R. Oliveira, Mar 08 2024: (Start)
a(n) = 2*a(n-1) - a(n-2) for n >= 2.
E.g.f.: 2*exp(x)*(1 + 3*x). (End)

A008611 a(n) = a(n-3) + 1, with a(0)=a(2)=1, a(1)=0.

Original entry on oeis.org

1, 0, 1, 2, 1, 2, 3, 2, 3, 4, 3, 4, 5, 4, 5, 6, 5, 6, 7, 6, 7, 8, 7, 8, 9, 8, 9, 10, 9, 10, 11, 10, 11, 12, 11, 12, 13, 12, 13, 14, 13, 14, 15, 14, 15, 16, 15, 16, 17, 16, 17, 18, 17, 18, 19, 18, 19, 20, 19, 20, 21, 20, 21, 22, 21, 22, 23, 22, 23, 24, 23, 24, 25, 24, 25, 26, 25, 26, 27, 26, 27, 28

Offset: 0

N. J. A. Sloane, Mar 15 1996

Molien series of 2-dimensional representation of cyclic group of order 3 over GF(2).
One step back, two steps forward.
The crossing number of the graph C(n, {1,3}), n >= 8, is [n/3] + n mod 3, which gives this sequence starting at the first 4. [Yang Yuansheng et al.]
A Chebyshev transform of A078008. The g.f. is the image of (1-x)/(1-x-2*x^2) (g.f. of A078008) under the Chebyshev transform A(x)-> (1/(1+x^2))*A(x/(1+x^2)). - Paul Barry, Oct 15 2004
A047878 is an essentially identical sequence. - Anton Chupin, Oct 24 2009
Rhyme scheme of Dante Alighieri's "Divine Comedy." - David Gaita, Feb 11 2011
A194960 results from deleting the first four terms of A008611. Note that deleting the first term or first four terms of A008611 leaves a concatenation of segments (n, n+1, n+2); for related concatenations, see
  A008619, (n,n+1) after deletion of first term;
  A053737, (n,n+1,n+2,n+3) beginning with n=0;
  A053824, (n to n+4) beginning with n=0. - Clark Kimberling, Sep 07 2011
It appears that a(n) is the number of roots of x^(n+1) + x + 1 inside the unit circle. - Michel Lagneau, Nov 02 2012
Also apparently for n >= 2: a(n) is the largest remainder r that results from dividing n+2 by 1..n+2 more than once, i.e., a(n) = max(i, A072528(n+2,i)>1). - Ralf Stephan, Oct 21 2013
Number of n-element subsets of [n+1] whose sum is a multiple of 3. a(4) = 1: {1,2,4,5}. - Alois P. Heinz, Feb 06 2017
It appears that a(n) is the number of roots of the Fibonacci polynomial F(n+2,x) strictly inside the unit circle of the complex plane. - Michel Lagneau, Apr 07 2017
For the proof of the preceding conjecture see my comments under A008615 and A049310. Chebyshev S(n,x) = i^n*F(n+1,-i*x), with i = sqrt(-1). - Wolfdieter Lang, May 06 2017
The sequence is the interleaving of three sequences: the positive integers (A000027), the nonnegative integers (A001477), and the positive integers, in that order. - Guenther Schrack, Nov 07 2020
a(n) is the number of multiples of 3 between n and 2n. - Christian Barrientos, Dec 20 2021
a(n) is the least number of football games a team has to play to be able to get n-1 points, where a win is 3 points, a draw is 1 point, and a loss is 0 points. - Sigurd Kittilsen, Dec 01 2022

			G.f. = 1 + x^2 + 2*x^3 + x^4 + 2*x^5 + 3*x^6 + 2*x^7 + 3*x^8 + 4*x^9 + ...

D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 103.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Cristian Cobeli, Aaditya Raghavan, and Alexandru Zaharescu, On the central ball in a translation invariant involutive field, arXiv:2408.01864 [math.NT], 2024. See p. 7.
Frederik Glitzner and David Manlove, Perspectives on Unsolvability in Roommates Markets, arXiv:2505.06717 [cs.GT], 2025. See p. 13.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 447.
Gerard P. Michon, Counting Polyhedra.
Yang Yuansheng et al., The crossing number of C(n; {1,3}), Discr. Math. 289 (2004), 107-118.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).
Index entries for Molien series.

Cf. A000027, A001477, A008619, A047878, A049310, A049347, A053737, A053824, A058207, A058788, A072528, A078008, A194960, A257075, A330396.

a008611 n = n' + mod r 2 where (n', r) = divMod (n + 1) 3
a008611_list = f [1,0,1] where f xs = xs ++ f (map (+ 1) xs)
-- Reinhard Zumkeller, Nov 25 2013

[(n-1)-2*Floor((n-1)/3): n in [0..90]]; // Vincenzo Librandi, Aug 21 2011

with(numtheory): for n from 1 to 70 do:it:=0:
y:=[fsolve(x^n+x+1, x, complex)] : for m from 1 to nops(y) do : if abs(y[m])< 1 then it:=it+1:else fi:od: printf(`%d, `,it):od:
A008611:=n->(n-1)-2*floor((n-1)/3); seq(A008611(n), n=0..50); # Wesley Ivan Hurt, May 18 2014

With[{nn=30},Riffle[Riffle[Range[nn],Range[0,nn-1]],Range[nn],3]] (* or *) RecurrenceTable[{a[0]==a[2]==1,a[1]==0,a[n]==a[n-3]+1},a,{n,90}] (* Harvey P. Dale, Nov 06 2011 *)
LinearRecurrence[{1, 0, 1, -1}, {1, 0, 1, 2}, 100] (* Vladimir Joseph Stephan Orlovsky, Feb 23 2012 *)
a[ n_] := Quotient[n - 1, 3] + Mod[n + 2, 3]; (* Michael Somos, Jan 23 2014 *)

{a(n) = (n-1) \ 3 + (n+2) % 3}; /* Michael Somos, Jan 23 2014 */

a(n) = a(n-3) + 1.
a(n) = (n-1) - 2*floor((n-1)/3).
G.f.: (1 + x^2 + x^4)/(1 - x^3)^2.
After the initial term, has form {n, n+1, n+2} for n=0, 1, 2, ...
From Paul Barry, Mar 18 2004: (Start)
a(n) = Sum_{k=0..n} (-1)^floor(2*(k-2)/3);
a(n) = 4*sqrt(3)*cos(2*Pi*n/3 + Pi/6)/9 + (n+1)/3. (End)
From Paul Barry, Oct 15 2004: (Start)
G.f.: (1 - x + x^2)/((1 + x + x^2)*(x-1)^2);
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*A078008(n-2k)*(-1)^k. (End)
a(n) = -a(-2-n) for all n in Z.
Euler transform of length 6 sequence [0, 1, 2, 0, 0, -1]. - Michael Somos, Jan 23 2014
a(n) = ((n-1) mod 3) + floor((n-1)/3). - Wesley Ivan Hurt, May 18 2014
PSUM transform of A257075. - Michael Somos, Apr 15 2015
a(n) = A194960(n-3), n >= 0, with extended A194960. See the a(n) formula two lines above. - Wolfdieter Lang, May 06 2017
From Guenther Schrack, Nov 07 2020: (Start)
a(n) = (3*n + 3 + 2*(w^(2*n)*(1 - w) + w^n*(2 + w)))/9, where w = (-1 + sqrt(-3))/2, a primitive third root of unity;
a(n) = (n + 1 + 2*A049347(n))/3;
a(n) = (2*n - A330396(n-1))/3. (End)
E.g.f.: (3*exp(x)*(1 + x) + exp(-x/2)*(6*cos(sqrt(3)*x/2) - 2*sqrt(3)*sin(sqrt(3)*x/2)))/9. - Stefano Spezia, May 06 2022
Sum_{n>=2} (-1)^n/a(n) = 3*log(2) - 1. - Amiram Eldar, Sep 10 2023

A103221 Number of partitions of n into parts 2 and 3.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 3, 3, 3, 3, 4, 3, 4, 4, 4, 4, 5, 4, 5, 5, 5, 5, 6, 5, 6, 6, 6, 6, 7, 6, 7, 7, 7, 7, 8, 7, 8, 8, 8, 8, 9, 8, 9, 9, 9, 9, 10, 9, 10, 10, 10, 10, 11, 10, 11, 11, 11, 11, 12, 11, 12, 12, 12, 12, 13, 12, 13, 13, 13, 13, 14, 13, 14, 14, 14, 14, 15, 14, 15, 15

Offset: 0

Michael Somos, Jan 25 2005

Essentially the same as A008615.
Poincaré series [or Poincare series] for modular forms of weight w for the full modular group. As generators one may take the Eisenstein series E_4 (A004009) and E_6 (A013973).
Dimension of the space of weight 2n+12 cusp forms for Gamma_0( 1 ).
Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 5 ).
a(n) is the number of partitions of n into two nonnegative parts congruent modulo 3. - Andrew Baxter, Jun 28 2006
Also number of equivalence classes of period 2n billiards on an equilateral triangle. - Andrew Baxter, Jun 06 2008
a(n) is also the number of 2-regular multigraphs on n vertices, where each component is either a pair of parallel edges, or a triangle. - Jason Kimberley, Oct 14 2011
For n>1, a(n) is the number of partitions of 2n into positive parts x,y, and z such that x>=y and y=z. This sequence is used in calculating the probability of the need for a run-off election when n voters randomly cast ballots for two of three candidates running for two empty slots on a county commission. - Dennis P. Walsh, Apr 25 2013
Also, Molien series for invariants of finite Coxeter group A_2. The Molien series for the finite Coxeter group of type A_k (k >= 1) has g.f. = 1/Product_{i=2..k+1} (1-x^i). Note that this is the root system A_k, not the alternating group Alt_k. - N. J. A. Sloane, Jan 11 2016
The coefficient of x^(2*n+1) in the power series expansion of the Weierstrass sigma function is a polynomial in the invariants g2 and g3 with a(n) terms. - Michael Somos, Jun 14 2016
a(n) is also the dimension of the complex vector space of modular forms M_{2*n} of weight 2*n and level 1 (full modular group). See Apostol p. 119, eq. (9) for k=2*n, and Ash and Gross, p. 178, Table 13.1. For a(6*k+1) = a(6*k+j)-1 for j = 0,2,3,4,5 and k >= 0 see A016921 (so-called dips, cf. Ash and Gross, p. 178.). - Wolfdieter Lang, Sep 16 2016
In an hexagonal tiling of the plane where the base tile is (0,0)--(2,1)--(3,3)--(1,4)--(-1,3)--(-2,1)--(0,0), a(n) is the number of vertices on the (n,0)--(n,n) closed line segment. - Luc Rousseau, Mar 22 2018

			For n=8, a(n)=2 since there are two partitions of 16 into 3 positive parts x, y, and z such that x >= y and y=z, namely, 16 = 8+4+4 and 16 = 6+5+5. - _Dennis P. Walsh_, Apr 25 2013
G.f. = 1 + x^2 + x^3 + x^4 + x^5 + 2*x^6 + x^7 + 2*x^8 + 2*x^9 + 2*x^10 + 2*x^11 + ...

T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, 1990, page 119.
Avner Ash and Robert Gross, Summing it up, Princeton University Press, 2016, p. 178.
D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 100.
E. Freitag, Siegelsche Modulfunktionen, Springer-Verlag, Berlin, 1983; p. 141, Th. 1.1.
R. C. Gunning, Lectures on Modular Forms. Princeton Univ. Press, Princeton, NJ, 1962.
J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.
J.-M. Kantor, Ou en sont les mathématiques, La formule de Molien-Weyl, SMF, Vuibert, p. 79
S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 26. - N. J. A. Sloane, Aug 28 2010.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Max A. Alekseyev and Allan Bickle, Forbidden Subgraphs of Single Graphs, (2024). See p. 5.
Andrew M. Baxter and Ron Umble, Periodic Orbits of Billiards on an Equilateral Triangle, Amer. Math. Monthly, 115 (No. 6, 2008), 479-491.
Ansgar Freyer, Monika Ludwig, and Martin Rubey, Unimodular Valuations beyond Ehrhart, arXiv:2407.07691 [math.MG], 2024. See p. 8.
Jun-Ichi Igusa, On Siegel modular forms of genus 2 (II), Amer. J. Math., 86 (1964), 392-412, esp. p. 402.
Mohammed L. Nadji, Moussa Ahmia, Daniel F. Checa, and José L. Ramírez, Arndt Compositions with Restricted Parts, Palindromes, and Colored Variants, J. Int. Seq. (2025) Vol. 28, Issue 3, Article 25.3.6. See p. 15.
Luc Rousseau, a(n) in an hexagonal tiling
Tetsuji Shioda, On the graded ring of invariants of binary octavics, Amer. J. Math. 89, 1022-1046, 1967.
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N))
William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_0(N))
William A. Stein, The modular forms database
Index entries for Molien series
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,-1).

Cf. A008615, A001399 (partial sums), A128115, A171386, A081753.
Molien series for finite Coxeter groups A_1 through A_12 are A059841, A103221, A266755, A008667, A037145, A001996, and A266776, A266777, A266778, A266779, A266780, A266781.
Cf. A004009, A013973, A016921, A079978.

[Floor((n+2)/2)-Floor((n+2)/3): n in [0..100]]; // Vincenzo Librandi, Sep 18 2016

A103221:=n->floor((n+2)/2)-floor((n+2)/3): # Andrew Baxter, Jun 06 2008

a=b=c=d=0;Table[e=a+b-d+1;a=b;b=c;c=d;d=e,{n,100}] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2011 *)
LinearRecurrence[{0, 1, 1, 0, -1},{1, 0, 1, 1, 1},88] (* Ray Chandler, Sep 23 2015 *)
a[ n_] := With[{m = Max[-5 - n, n]}, (-1)^Boole[n < 0] SeriesCoefficient[ 1 / ((1 - x^2) (1 - x^3)), {x, 0, m}]]; (* Michael Somos, Jun 02 2019 *)

{a(n) = if( n<-4, -a(-5-n), polcoeff( 1 / ((1 - x^2) * (1 - x^3)) + x * O(x^n), n))};

a(n)=n+=2; n\2 - n\3 \\ Charles R Greathouse IV, Jul 31 2017

def A103221(n): return (n>>1)+1-(n+2)//3 # Chai Wah Wu, Apr 15 2025

def a(n) : return( len( CuspForms( Gamma0( 1), 2*n + 12, prec=1). basis())); # Michael Somos, May 29 2013

Euler transform of finite sequence [0, 1, 1] with offset 1, which is A171386.
a(n) = A008615(n+2). First differences of A001399.
a(n) = a(n-6) + 1 = a(n-2) + a(n-3) - a(n-5). - Henry Bottomley, Sep 02 2000
G.f.: 1/((1-x^2)*(1-x^3)).
a(n) = floor((n+2)/2) - floor((n+2)/3). - Andrew Baxter, Jun 06 2008
For odd n, a(n)=floor((n+3)/6). For even n, a(n)=floor((n+6)/6). - Dennis P. Walsh, Apr 25 2013
a(n) = floor(n/6)+1 unless n == 1 (mod 6); if n == 1 (mod 6), a(n) = floor(n/6). - Bob Selcoe, Sep 27 2014
a(n) = A081753(2*n); see the Dennis P. Walsh formula. - Wolfdieter Lang, Sep 16 2016
a(n)-a(n-2) = A079978(n). - R. J. Mathar, Jun 23 2021
E.g.f.: (3*(4 + x)*cosh(x) + exp(-x/2)*(6*cos(sqrt(3)*x/2) - 2*sqrt(3)*sin(sqrt(3)*x/2)) + 3*(1 + x)*sinh(x))/18. - Stefano Spezia, Mar 05 2023
a(n) = A008615(n-1)+A059841(n). - R. J. Mathar, May 03 2023

Name changed by Wolfdieter Lang, Sep 16 2016

A211540 Number of ordered triples (w,x,y) with all terms in {1..n} and 2w = 3x + 4y.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 19, 21, 24, 27, 30, 33, 37, 40, 44, 48, 52, 56, 61, 65, 70, 75, 80, 85, 91, 96, 102, 108, 114, 120, 127, 133, 140, 147, 154, 161, 169, 176, 184, 192, 200, 208, 217, 225, 234, 243, 252, 261, 271, 280, 290

Offset: 0

Clark Kimberling, Apr 15 2012

For a guide to related sequences, see A211422.
Also the number of partitions of n+1 into three parts, where each part > 1. - Peter Woodward, May 25 2015
a(n) is also equal to the number of partitions of n+4 into three distinct parts, where each part > 1. - Giovanni Resta, May 26 2015
Number of different distributions of n+1 identical balls in 3 boxes as x,y,z where 0 < x < y < z. - Ece Uslu and Esin Becenen, Dec 31 2015
After the first three terms, partial sums of A008615. - Robert Israel, Dec 31 2015
For n >= 2, also the number of partitions of n - 2 into 3 parts. The Heinz numbers of these partitions are given by A014612. - Gus Wiseman, Oct 11 2020

			a(5) = a(6) = 1 with only one ordered triple (5,2,1). - _Michael Somos_, Feb 02 2015
a(11) = 5 Number of different distributions of 11 identical balls in 3 boxes as x,y and z where 0 < x < y < z. - _Ece Uslu_, Esin Becenen, Dec 31 2015
a(1) = a(2) = a(3) = a(4) = a(5) = 0, since with fewer than 6 identical balls there is no such distribution with 3 boxes that holds for 0 < x < y < z. - _Ece Uslu_, Esin Becenen, Dec 31 2015
G.f.: x^5 + x^6 + 2*x^7 + 3*x^8 + 4*x^9 + 5*x^10 + 7*x^11 + 8*x^12 + ...
From _Gus Wiseman_, Oct 11 2020: (Start)
The a(5) = 1 through a(15) = 14 partitions of n + 1 into three parts > 1 [Woodward] are the following (A = 10, B = 11, C = 12). The ordered version is A000217(n - 4) and the Heinz numbers are A046316.
  222  322  332  333  433  443  444  544  554  555  655
            422  432  442  533  543  553  644  654  664
                 522  532  542  552  643  653  663  754
                      622  632  633  652  662  744  763
                           722  642  733  743  753  772
                                732  742  752  762  844
                                822  832  833  843  853
                                     922  842  852  862
                                          932  933  943
                                          A22  942  952
                                               A32  A33
                                               B22  A42
                                                    B32
                                                    C22
The a(5) = 1 through a(15) = 14 partitions of n + 4 into three distinct parts > 1 [Resta] are the following (A = 10, B = 11, C = 12, D = 13, E = 14). The ordered version is A211540*6 and the Heinz numbers are A046389.
  432  532  542  543  643  653  654  754  764  765  865
            632  642  652  743  753  763  854  864  874
                 732  742  752  762  853  863  873  964
                      832  842  843  862  872  954  973
                           932  852  943  953  963  982
                                942  952  962  972  A54
                                A32  A42  A43  A53  A63
                                     B32  A52  A62  A72
                                          B42  B43  B53
                                          C32  B52  B62
                                               C42  C43
                                               D32  C52
                                                    D42
                                                    E32
The a(5) = 1 through a(15) = 14 partitions of n + 1 into three distinct parts [Uslu and Becenen] are the following (A = 10, B = 11, C = 12, D = 13). The ordered version is A211540(n)*6 and the Heinz numbers are A007304.
  321  421  431  432  532  542  543  643  653  654  754
            521  531  541  632  642  652  743  753  763
                 621  631  641  651  742  752  762  853
                      721  731  732  751  761  843  862
                           821  741  832  842  852  871
                                831  841  851  861  943
                                921  931  932  942  952
                                     A21  941  951  961
                                          A31  A32  A42
                                          B21  A41  A51
                                               B31  B32
                                               C21  B41
                                                    C31
                                                    D21
(End)

Robert Israel, Table of n, a(n) for n = 0..10000
Clark Kimberling, A Combinatorial Classification of Triangle Centers on the Line at Infinity, J. Int. Seq., Vol. 22 (2019), Article 19.5.4.
Ece Uslu and Esin Becenen, Identical Object Distributions.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Cf. A001399, A069905, A211422.
All of the following pertain to 3-part strict partitions.
- A000009 counts these partitions of any length, with non-strict version A000041.
- A007304 gives the Heinz numbers, with non-strict version A014612.
- A101271 counts the relatively prime case, with non-strict version A023023.
- A220377 counts the pairwise coprime case, with non-strict version A307719.
- A337605 counts the pairwise non-coprime case, with non-strict version A337599.
Cf. A000217, A001840, A156040, A284825, A337453, A337483, A337484, A337563.

I:=[0,0,0,0,0,1]; [n le 6 select I[n] else Self(n-1)+Self(n-2)-Self(n-4)-Self(n-5)+Self(n-6): n in [1..70]]; // Vincenzo Librandi, Dec 31 2015

f:= gfun:-rectoproc({a(n) = a(n-1)+a(n-2)-a(n-4)-a(n-5)+a(n-6),seq(a(i)=0,i=0..4),a(5)=1},a(n),remember):
seq(f(i),i=0..100); # Robert Israel, Dec 31 2015

t[n_] := t[n] = Flatten[Table[-2 w + 3 x + 4 y, {w, n}, {x, n}, {y, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 80}]  (* A211540 *)
FindLinearRecurrence[t]
LinearRecurrence[{1, 1, 0, -1, -1, 1}, {0, 0, 0, 0, 0, 1}, 70] (* Vincenzo Librandi, Dec 31 2015 *)
Table[Length[Select[IntegerPartitions[n+1,{3}],UnsameQ@@#&]],{n,0,30}] (* Gus Wiseman, Oct 05 2020 *)

{a(n) = round( (n-2)^2 / 12 )}; / * Michael Somos, Feb 02 2015 */

concat(vector(5), Vec(x^5/(1-x-x^2+x^4+x^5-x^6) + O(x^100))) \\ Altug Alkan, Jan 10 2016

a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6).
a(n) = A069905(n-2) = A001399(n-5) for n >= 5. - Alois P. Heinz, Nov 03 2012
a(n) = 3*k^2-6*k+3 (for n = 6*k-3), 3*k^2-5*k+2 (for n = 6*k-2), 3*k^2-4*k+1 (for n = 6*k-1), 3*k^2-3*k+1 (for n = 6*k), 3*k^2-2*k (for n = 6*k+1), 3*k^2-k (for n = 6*k+2). - Ece Uslu, Esin Becenen, Dec 31 2015
a(n) = A004526(n-2) + a(n-2) for n > 2. - Ece Uslu, Esin Becenen, Dec 31 2015
G.f.: x^5/(1 - x - x^2 + x^4 + x^5 - x^6). - Robert Israel, Dec 31 2015
a(n) = Sum_{k=1..floor(n/3)} floor((n-k)/2)-k. - Wesley Ivan Hurt, Apr 27 2019
From Gus Wiseman, Oct 11 2020: (Start)
a(n+2) = A069905(n) = A001399(n-3) counts 3-part partitions.
a(n-1) = A069905(n-3) = A001399(n-6) counts 3-part strict partitions.
a(n-1) = A069905(n-3) = A001399(n-6) counts 3-part partitions with no 1's.
a(n-4) = A069905(n-6) = A001399(n-9) counts 3-part strict partitions with no 1's.
A000217(n-2) counts 3-part compositions.
a(n-1)*6 = A069905(n-3)*6 = A001399(n-6)*6 counts 3-part strict compositions.
A000217(n-5) counts 3-part compositions with no 1's.
a(n-4)*6 = A069905(n-6)*6 = A001399(n-9)*6 counts 3-part strict compositions with no 1's.
(End)

cp's OEIS Frontend

A016969 a(n) = 6*n + 5.

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A016957 a(n) = 6*n + 4.

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A016933 a(n) = 6*n + 2.

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A008611 a(n) = a(n-3) + 1, with a(0)=a(2)=1, a(1)=0.

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A103221 Number of partitions of n into parts 2 and 3.

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