A035361 -id:A035361 - cp's OEIS frontend

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

1, 1, 2, 2, 4, 5, 7, 9, 13, 16, 22, 27, 36, 44, 57, 70, 89, 108, 135, 163, 202, 243, 297, 355, 431, 513, 617, 731, 874, 1031, 1225, 1439, 1701, 1991, 2341, 2731, 3197, 3717, 4333, 5022, 5834, 6741, 7803, 8991, 10375, 11923, 13716, 15723, 18038, 20628, 23603

Offset: 0

N. J. A. Sloane

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 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
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
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
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
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

			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. 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.
Riccardo Aragona, Roberto Civino, and Norberto Gavioli, A modular idealizer chain and unrefinability of partitions with repeated parts, arXiv:2301.06347 [math.RA], 2023.
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.
Wikipedia, Glaisher's Theorem.

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.

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

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 *)
Table[Count[IntegerPartitions[n],?(NoneTrue[Mod[#,3]==0&])],{n,0,50}] (* _Harvey P. Dale, Sep 06 2022 *)

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

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

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) = Product_{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

More terms from Olivier Gérard

A007494 Numbers that are congruent to 0 or 2 mod 3.

Original entry on oeis.org

0, 2, 3, 5, 6, 8, 9, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 26, 27, 29, 30, 32, 33, 35, 36, 38, 39, 41, 42, 44, 45, 47, 48, 50, 51, 53, 54, 56, 57, 59, 60, 62, 63, 65, 66, 68, 69, 71, 72, 74, 75, 77, 78, 80, 81, 83, 84, 86, 87, 89, 90, 92, 93, 95, 96, 98, 99, 101, 102, 104, 105, 107

Offset: 0

Christopher Lam Cham Kee (Topher(AT)CyberDude.Com)

The map n -> a(n) (where a(n) = 3n/2 if n even or (3n+1)/2 if n odd) was studied by Mahler, in connection with "Z-numbers" and later by Flatto. One question was whether, iterating from an initial integer, one eventually encountered an iterate = 1 (mod 4). - Jeff Lagarias, Sep 23 2002
Partial sums of 0,2,1,2,1,2,1,2,1,... . - Paul Barry, Aug 18 2007
a(n) = numbers k such that antiharmonic mean of the first k positive integers is not integer. A169609(a(n-1)) = 3. See A146535 and A169609. Complement of A016777. - Jaroslav Krizek, May 28 2010
Range of A173732. - Reinhard Zumkeller, Apr 29 2012
Number of partitions of 6n into two odd parts. - Wesley Ivan Hurt, Nov 15 2014
Numbers m such that 3 divides A000217(m). - Bruno Berselli, Aug 04 2017
Maximal length of a snake like polyomino that fits in a 2 X n rectangle. - Alain Goupil, Feb 12 2020

L. Flatto, Z-numbers and beta-transformations, in Symbolic dynamics and its applications (New Haven, CT, 1991), 181-201, Contemp. Math., 135, Amer. Math. Soc., Providence, RI, 1992.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1002.
Attila Máder, The Use of Experimental Mathematics in the Classroom, in Interesting Mathematical Problems in Sciences and Everyday Life - 2011.
Kurt Mahler, An unsolved problem on the powers of 3/2, J. Austral. Math. Soc., Vol. 8 (1968), pp. 313-321.
P. Sabinin and M. G. Stone, Transforming n-gons by Folding the Plane, Amer. Math. Monthly, Vol. 102, No. 7 (1995), pp. 620-627.
Eric Weisstein's World of Mathematics, Folding.
Robert G. Wilson v, Notes with attachment.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A000217, A001651, A032766, A035361, A063574, A132462.
Complement of A016777.
Range of A002517.
Cf. A274406. [Bruno Berselli, Jun 26 2016]

a007494 =  flip div 2 . (+ 1) . (* 3) -- Reinhard Zumkeller, Dec 12 2014

[(6*n+1)/4-(-1)^n/4: n in [0..80]]; // Vincenzo Librandi, Aug 20 2011

a[0]:=0:a[1]:=2:for n from 2 to 100 do a[n]:=a[n-2]+3 od: seq(a[n], n=0..71); # Zerinvary Lajos, Mar 16 2008
A007494:=n->floor((3*n+1)/2); seq(A007494(k), k=0..100); # Wesley Ivan Hurt, Sep 27 2013

Flatten[{#,#+2}&/@(3Range[0,40])] (* Harvey P. Dale, May 15 2011 *)
Table[2n - Floor[n/2], {n,0,100}] (* Wesley Ivan Hurt, Sep 27 2013 *)

a(n)=n+(n+1)>>1 \\ Charles R Greathouse IV, Jul 25 2011

a(n) = 3*n/2 if n even, otherwise (3*n+1)/2.
If u(1)=0, u(n) = n + floor(u(n-1)/3), then a(n-1) = u(n). - Benoit Cloitre, Nov 26 2002
G.f.: x*(x+2)/((1-x)^2*(1+x)). - Ralf Stephan, Apr 13 2002
a(n) = 3*floor(n/2) + 2*(n mod 2) = A032766(n) + A000035(n). - Reinhard Zumkeller, Apr 04 2005
a(n) = (6*n+1)/4 - (-1)^n/4; a(n) = Sum_{k=0..n-1} (1 + (-1)^(k/2)*cos(k*Pi/2)). - Paul Barry, Aug 18 2007
A145389(a(n)) <> 1. - Reinhard Zumkeller, Oct 10 2008
a(n) = A002943(n) - A173511(n). - Reinhard Zumkeller, Feb 20 2010
a(n) = 3*n - a(n-1) - 1 (with a(0)=0). - Vincenzo Librandi, Nov 18 2010
a(n) = Sum_{k>=0} A030308(n,k)*A042950(k). - Philippe Deléham, Oct 17 2011
a(n) = n + ceiling(n/2). - Arkadiusz Wesolowski, Sep 18 2012
a(n) = 2n - floor(n/2) = floor((3n+1)/2) = n + (n + (n mod 2))/2. - Wesley Ivan Hurt, Oct 19 2013
a(n) = A000217(n+1) - A099392(n+1). - Bui Quang Tuan, Mar 27 2015
a(n) = n + floor(n/2) + (n mod 2). - Bruno Berselli, Apr 04 2016
a(n) = Sum_{i=1..n} numerator(2/i). - Wesley Ivan Hurt, Feb 26 2017
a(n) = Sum_{k=0..n-1} Sum_{i=0..k} C(i,k)+(-1)^(k-i). - Wesley Ivan Hurt, Sep 20 2017
E.g.f.: (3*exp(x)*x + sinh(x))/2. - Stefano Spezia, Feb 11 2020
Sum_{n>=1} (-1)^(n+1)/a(n) = log(3)/2 - Pi/(6*sqrt(3)). - Amiram Eldar, Dec 04 2021

A132462 Number of partitions of n into distinct parts congruent to 0 or 2 modulo 3.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 1, 1, 3, 2, 2, 5, 2, 4, 7, 4, 7, 10, 6, 11, 14, 9, 17, 19, 14, 25, 26, 21, 36, 35, 31, 50, 47, 45, 69, 63, 64, 93, 84, 89, 125, 111, 124, 165, 147, 169, 216, 194, 227, 281, 254, 303, 363, 332, 400, 466, 432, 523, 595, 559, 680, 756, 721, 876, 956, 926, 1121

Offset: 0

Reinhard Zumkeller, Aug 22 2007

nonn

			a(8)=3 because we have 8, 6+2 and 5+3.

Reinhard Zumkeller, Table of n, a(n) for n = 0..200

Cf. A007494, A035361, A003105, A132463.

g:=product((1+x^(3*k))*(1+x^(3*k-1)),k=1..30): gser:=series(g,x=0,100): seq(coeff(gser,x,n),n=0..70); # Emeric Deutsch, Aug 30 2007

nmax = 40; CoefficientList[Series[Product[((1+x^(3*k))*(1+x^(3*k-1))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 24 2015 *)

G.f.: Product_{k>=1} (1+x^(3*k))*(1+x^(3*k-1)). - Emeric Deutsch, Aug 30 2007

a(n) ~ exp(Pi*sqrt(2*n)/3) / (2^(23/12) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Aug 24 2015

a(0)=1 prepended by Vaclav Kotesovec, Aug 24 2015

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

Formula

Extensions

A007494 Numbers that are congruent to 0 or 2 mod 3.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Formula

A132462 Number of partitions of n into distinct parts congruent to 0 or 2 modulo 3.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A035360 Number of partitions of n into parts 3k or 3k+1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A374060 Expansion of Product_{k>=1} (1 - x^(3*k-1)) * (1 - x^(3*k)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A374060 Expansion of Product_{k>=1} (1 - x^(3k-1)) (1 - x^(3*k)).