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A000726 Number of partitions of n in which no parts are multiples of 3.

%I A000726 M0316 N0116 #123 Feb 16 2025 08:32:21
%S A000726 1,1,2,2,4,5,7,9,13,16,22,27,36,44,57,70,89,108,135,163,202,243,297,
%T A000726 355,431,513,617,731,874,1031,1225,1439,1701,1991,2341,2731,3197,3717,
%U A000726 4333,5022,5834,6741,7803,8991,10375,11923,13716,15723,18038,20628,23603
%N A000726 Number of partitions of n in which no parts are multiples of 3.
%C A000726 Case k=4, i=3 of Gordon Theorem.
%C A000726 Expansion of q^(-1/12)*eta(q^3)/eta(q) in powers of q. - _Michael Somos_, Apr 20 2004
%C A000726 Euler transform of period 3 sequence [1,1,0,...]. - _Michael Somos_, Apr 20 2004
%C A000726 Also the 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
%C A000726 Also the number of partitions of n where no part 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
%C A000726 Also the 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
%C A000726 Equals left border of triangle A174714. - _Gary W. Adamson_, Mar 27 2010
%C A000726 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
%C A000726 Convolution of A035382 and A035386. - _Vaclav Kotesovec_, Aug 23 2015
%C A000726 The number of partitions of n in which no parts are multiples of k equals the number of partitions of n where no part appears more than k-1 times. - _Gregory L. Simay_, Oct 15 2022
%D A000726 G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109.
%D A000726 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.
%D A000726 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000726 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000726 T. D. Noe and Vaclav Kotesovec, <a href="/A000726/b000726.txt">Table of n, a(n) for n = 0..5000</a> (terms 0..1000 from T. D. Noe)
%H A000726 George E. Andrews, <a href="https://hal.archives-ouvertes.fr/hal-03498190/">Partition Identities for Two-Color Partitions</a>, 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.
%H A000726 Riccardo Aragona, Roberto Civino, and Norberto Gavioli, <a href="https://arxiv.org/abs/2301.06347">A modular idealizer chain and unrefinability of partitions with repeated parts</a>, arXiv:2301.06347 [math.RA], 2023.
%H A000726 N. Chair, <a href="http://arXiv.org/abs/hep-th/0409011">Partition identities from Partial Supersymmetry</a>, arXiv:hep-th/0409011, 2004.
%H A000726 Edray Herber Goins and Talitha M. Washington, <a href="https://arxiv.org/abs/0909.5459">On the generalized climbing stairs problem</a>, Ars Combin.  117  (2014), 183-190.  MR3243840 (Reviewed), arXiv:0909.5459 [math.CO], 2009.
%H A000726 Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 15.
%H A000726 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PartitionFunctionb.html">Partition function b_k.</a>
%H A000726 Wikipedia, <a href="https://en.wikipedia.org/wiki/Glaisher%27s_theorem">Glaisher's Theorem</a>.
%F A000726 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).
%F A000726 a(n) = A061197(n, n).
%F A000726 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
%F A000726 G.f.: P(x^3)/P(x) where P(x) = Product_{k>=1} (1 - x^k). - _Joerg Arndt_, Jun 21 2011
%F A000726 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
%F A000726 a(n) = (1/n)*Sum_{k=1..n} A046913(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, Mar 21 2017
%F A000726 G.f.: exp(Sum_{k>=1} x^k*(1 + x^k)/(k*(1 - x^(3*k)))). - _Ilya Gutkovskiy_, Aug 15 2018
%e A000726 There are a(6)=7 partitions of 6 into parts != 0 (mod 3):
%e A000726 [ 1]  [5,1],
%e A000726 [ 2]  [4,2],
%e A000726 [ 3]  [4,1,1],
%e A000726 [ 4]  [2,2,2],
%e A000726 [ 5]  [2,2,1,1],
%e A000726 [ 6]  [2,1,1,1,1], and
%e A000726 [ 7]  [1,1,1,1,1,1]
%e A000726 .
%e A000726 From _Joerg Arndt_, Dec 29 2012: (Start)
%e A000726 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):
%e A000726 [ 1]  [ 3 3 2 1 1 ]
%e A000726 [ 2]  [ 3 3 2 2 ]
%e A000726 [ 3]  [ 4 2 2 1 1 ]
%e A000726 [ 4]  [ 4 3 2 1 ]
%e A000726 [ 5]  [ 4 3 3 ]
%e A000726 [ 6]  [ 4 4 1 1 ]
%e A000726 [ 7]  [ 4 4 2 ]
%e A000726 [ 8]  [ 5 2 2 1 ]
%e A000726 [ 9]  [ 5 3 1 1 ]
%e A000726 [10]  [ 5 3 2 ]
%e A000726 [11]  [ 5 4 1 ]
%e A000726 [12]  [ 5 5 ]
%e A000726 [13]  [ 6 2 1 1 ]
%e A000726 [14]  [ 6 2 2 ]
%e A000726 [15]  [ 6 3 1 ]
%e A000726 [16]  [ 6 4 ]
%e A000726 [17]  [ 7 2 1 ]
%e A000726 [18]  [ 7 3 ]
%e A000726 [19]  [ 8 1 1 ]
%e A000726 [20]  [ 8 2 ]
%e A000726 [21]  [ 9 1 ]
%e A000726 [22]  [ 10 ]
%e A000726 (End)
%p A000726 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
%p A000726 # second Maple program:
%p A000726 with(numtheory):
%p A000726 a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
%p A000726      `if`(irem(d, 3)=0, 0, d), d=divisors(j)), j=1..n)/n)
%p A000726     end:
%p A000726 seq(a(n), n=0..50);  # _Alois P. Heinz_, Nov 17 2017
%t A000726 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 *)
%t A000726 QP = QPochhammer; CoefficientList[QP[q^3]/QP[q] + O[q]^60, q] (* _Jean-François Alcover_, Nov 24 2015 *)
%t A000726 nmax = 50; CoefficientList[Series[Product[(1 - x^(3*k))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jan 02 2016 *)
%t A000726 Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 3], 0, 2] ], {n, 0, 50}] (* _Robert Price_, Jul 28 2020 *)
%t A000726 Table[Count[IntegerPartitions[n],_?(NoneTrue[Mod[#,3]==0&])],{n,0,50}] (* _Harvey P. Dale_, Sep 06 2022 *)
%o A000726 (PARI) a(n)=if(n<0,0,polcoeff(eta(x^3+x*O(x^n))/eta(x+x*O(x^n)),n))
%o A000726 (PARI) lista(nn) = {q='q+O('q^nn); Vec(eta(q^3)/eta(q))} \\ _Altug Alkan_, Mar 20 2018
%o A000726 (Haskell)
%o A000726 a000726 n = p a001651_list n where
%o A000726    p _  0 = 1
%o A000726    p ks'@(k:ks) m | m < k     = 0
%o A000726                   | otherwise = p ks' (m - k) + p ks m
%o A000726 -- _Reinhard Zumkeller_, Aug 23 2011
%Y A000726 Cf. A000009 (no multiples of 2), A001935 (no of 4), A035959 (no of 5), A219601 (no of 6), A035985, A001651, A003105, A035361, A035360.
%Y A000726 Cf. A174714. - _Gary W. Adamson_, Mar 27 2010
%Y A000726 Cf. A113685, A176202. - _Gary W. Adamson_, Apr 11 2010
%Y A000726 Cf. A046913.
%Y A000726 Column k=3 of A286653.
%Y A000726 Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.
%K A000726 nonn,easy,nice
%O A000726 0,3
%A A000726 _N. J. A. Sloane_
%E A000726 More terms from _Olivier Gérard_