|
|
A000726
|
|
Number of partitions of n in which no parts are multiples of 3.
(Formerly M0316 N0116)
|
|
33
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
COMMENTS
|
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 [From 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
|
|
REFERENCES
|
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).
|
|
LINKS
|
T. D. Noe, Table of n, a(n) for n=0..1000
N. Chair, Partition identities from Partial Supersymmetry
Eric Weisstein's World of Mathematics, Partition function b_k.
|
|
FORMULA
|
G.f.: 1/(prod(k>=1, (1-x^(3*k-1))*(1-x^(3*k-2)))) = prod(k>=1, 1+x^k+x^(2*k) ) (where 1+x+x^2 is 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) ~ exp(2*Pi*sqrt(n)/3) / (6*n^(3/4)). - Vaclav Kotesovec, Aug 23 2015
|
|
EXAMPLE
|
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)
|
|
MAPLE
|
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
|
|
MATHEMATICA
|
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 *)
|
|
PROG
|
(PARI) a(n)=if(n<0, 0, polcoeff(eta(x^3+x*O(x^n))/eta(x+x*O(x^n)), n))
(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
|
|
CROSSREFS
|
Cf. A000009, A001935, A035959, A219601, A035985, A001651, A003105, A035361, A035360.
Cf. A174714. [From Gary W. Adamson, Mar 27 2010]
Cf. A113685, A176202. [From Gary W. Adamson, Apr 11 2010]
Sequence in context: A166239 A058661 A094362 * A128663 A206557 A240508
Adjacent sequences: A000723 A000724 A000725 * A000727 A000728 A000729
|
|
KEYWORD
|
nonn,easy,nice
|
|
AUTHOR
|
N. J. A. Sloane.
|
|
EXTENSIONS
|
More terms from Olivier Gérard
|
|
STATUS
|
approved
|
|
|
|