0,3

Case k=4, i=3 of Gordon Theorem.

Expansion of q^(-1/12)*eta(q^3)/eta(q) in powers of q. - Michael Somos, Apr 20 2004

Euler transform of period 3 sequence [1,1,0,...]. - Michael Somos, Apr 20 2004

Also number of partitions with at most 2 parts of size 1 and all differences between parts at distance 3 are greater than 1. Example: a(6)=7 because we have [6],[5,1],[4,2],[4,1,1],[3,3],[3,2,1] and [2,2,2] (for example, [2,2,1,1] does not qualify because the difference between the first and the fourth parts is equal to 1). - Emeric Deutsch, Apr 18 2006

Also number of partitions of n where no positive integer appears more than twice. Example: a(6)=7 because we have [6],[5,1],[4,2],[4,1,1],[3,3],[3,2,1] and [2,2,1,1]. - Emeric Deutsch, Apr 18 2006

Also number of partitions of n with least part either 1 or 2 and with differences of consecutive parts at most 2. Example: a(6)=7 because we have [4,2], [3,2,1], [3,1,1,1], [2,2,2], [2,2,1,1], [2,1,1,1,1] and [1,1,1,1,1,1]. - Emeric Deutsch, Apr 18 2006

Equals left border of triangle A174714. - Gary W. Adamson, Mar 27 2010

Triangle A113685 is equivalent to p(x) = p(x^2) * A000009(x); given A000041(x) = p(x). Triangle A176202 is equivalent to p(x) = p(x^3) * A000726(x). - Gary W. Adamson, Apr 11 2010

Convolution of A035382 and A035386. - Vaclav Kotesovec, Aug 23 2015

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109.

L. Carlitz, Generating functions and partition problems, pp. 144-169 of A. L. Whiteman, ed., Theory of Numbers, Proc. Sympos. Pure Math., 8 (1965). Amer. Math. Soc., see p. 145.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

T. D. Noe and Vaclav Kotesovec, Table of n, a(n) for n = 0..5000 (terms 0..1000 from T. D. Noe)

George E. Andrews, Partition Identities for Two-Color Partitions, Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 2021, Special Commemorative volume in honour of Srinivasa Ramanujan, 2021, 44, pp.74-80. hal-03498190. See p. 79.

N. Chair, Partition identities from Partial Supersymmetry, arXiv:hep-th/0409011, 2004.

Edray Herber Goins and Talitha M. Washington, On the generalized climbing stairs problem, Ars Combin.  117  (2014), 183-190.  MR3243840 (Reviewed), arXiv:0909.5459 [math.CO], 2009.

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 15.

Eric Weisstein's World of Mathematics, Partition function b_k.

G.f.: 1/(Product_{k>=1} (1-x^(3*k-1))*(1-x^(3*k-2))) = Product_{k>=1} (1 + x^k + x^(2*k)) (where 1 + x + x^2 is the 3rd cyclotomic polynomial).

a(n) = A061197(n, n).

Given g.f. A(x) then B(x)=x*A(x^6)^2 satisfies 0=f(B(x), B(x^2), B(x^4)) where f(u,v,w)= +v^2 +v*w^2 -v*u^2 +3*u^2*w^2. - Michael Somos, May 28 2006

G.f.: P(x^3)/P(x) where P(x)=prod(k>=1, 1-x^k ). - Joerg Arndt, Jun 21 2011

a(n) ~ 2*Pi * BesselI(1, sqrt((12*n + 1)/3)*Pi/3) / (3*sqrt(12*n + 1)) ~ exp(2*Pi*sqrt(n)/3) / (6*n^(3/4)) * (1 + (Pi/36 - 9/(16*Pi))/sqrt(n) + (Pi^2/2592 - 135/(512*Pi^2) - 5/64)/n). - Vaclav Kotesovec, Aug 23 2015, extended Jan 13 2017

a(n) = (1/n)*Sum_{k=1..n} A046913(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 21 2017

G.f.: exp(Sum_{k>=1} x^k*(1 + x^k)/(k*(1 - x^(3*k)))). - Ilya Gutkovskiy, Aug 15 2018

There are a(6)=7 partitions of 6 into parts !=0 (mod 3):

[ 1]  [5,1],

[ 2]  [4,2],

[ 3]  [4,1,1],

[ 4]  [2,2,2],

[ 5]  [2,2,1,1],

[ 6]  [2,1,1,1,1], and

[ 7]  [1,1,1,1,1,1]

.

From Joerg Arndt, Dec 29 2012: (Start)

There are a(10)=22 partitions p(1)+p(2)+...+p(m)=10 such that p(k)!=p(k-2) (that is, no part appears more than twice):

[ 1]  [ 3 3 2 1 1 ]

[ 2]  [ 3 3 2 2 ]

[ 3]  [ 4 2 2 1 1 ]

[ 4]  [ 4 3 2 1 ]

[ 5]  [ 4 3 3 ]

[ 6]  [ 4 4 1 1 ]

[ 7]  [ 4 4 2 ]

[ 8]  [ 5 2 2 1 ]

[ 9]  [ 5 3 1 1 ]

[10]  [ 5 3 2 ]

[11]  [ 5 4 1 ]

[12]  [ 5 5 ]

[13]  [ 6 2 1 1 ]

[14]  [ 6 2 2 ]

[15]  [ 6 3 1 ]

[16]  [ 6 4 ]

[17]  [ 7 2 1 ]

[18]  [ 7 3 ]

[19]  [ 8 1 1 ]

[20]  [ 8 2 ]

[21]  [ 9 1 ]

[22]  [ 10 ]

(End)

g:=product(1+x^j+x^(2*j), j=1..60): gser:=series(g, x=0, 55): seq(coeff(gser, x, n), n=0..50); # Emeric Deutsch, Apr 18 2006

# second Maple program:

with(numtheory):

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(

     `if`(irem(d, 3)=0, 0, d), d=divisors(j)), j=1..n)/n)

    end:

seq(a(n), n=0..50);  # Alois P. Heinz, Nov 17 2017

f[0] = 1; f[n_] := Coefficient[Expand@ Product[1 + x^k + x^(2k), {k, n}], x^n]; Table[f@n, {n, 0, 40}] (* Robert G. Wilson v, Nov 10 2006 *)

QP = QPochhammer; CoefficientList[QP[q^3]/QP[q] + O[q]^60, q] (* Jean-François Alcover, Nov 24 2015 *)

nmax = 50; CoefficientList[Series[Product[(1 - x^(3*k))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 02 2016 *)

Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 3], 0, 2] ], {n, 0, 50}] (* Robert Price, Jul 28 2020 *)

(PARI) a(n)=if(n<0, 0, polcoeff(eta(x^3+x*O(x^n))/eta(x+x*O(x^n)), n))

(PARI) lista(nn) = {q='q+O('q^nn); Vec(eta(q^3)/eta(q))} \\ Altug Alkan, Mar 20 2018

(Haskell)

a000726 n = p a001651_list n where

   p _  0 = 1

   p ks'@(k:ks) m | m < k     = 0

                  | otherwise = p ks' (m - k) + p ks m

-- Reinhard Zumkeller, Aug 23 2011

Cf. A000009 (no multiples of 2), A001935 (no of 4), A035959 (no of 5), A219601 (no of 6), A035985, A001651, A003105, A035361, A035360.

Cf. A174714. - Gary W. Adamson, Mar 27 2010

Cf. A113685, A176202. - Gary W. Adamson, Apr 11 2010

Cf. A046913.

Column k=3 of A286653.

Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.

Sequence in context: A166239 A058661 A094362 * A128663 A206557 A240508

Adjacent sequences:  A000723 A000724 A000725 * A000727 A000728 A000729

nonn,easy,nice

N. J. A. Sloane

More terms from Olivier Gérard

approved