A008676 Expansion of 1/((1-x^3)*(1-x^5)).

1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 4, 3, 4, 4, 3, 4, 4, 4, 4, 4, 4, 4, 5, 4, 4, 5, 4, 5, 5, 4, 5, 5, 5, 5, 5, 5, 5, 6, 5, 5

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(n) gives the number of partitions of n using only the parts 3 and 5. e.g. a(25)=2: 5+5+5+5+5 and 5+5+3+3+3+3+3+3. - Andrew Baxter, Jun 20 2011

a(n) gives the number of partitions of n+8 involving both a 3 and a 5. e.g. a(25)=2 and we may write 33 as 5+5+5+5+5+5+3 and 5+5+5+3+3+3+3+3+3. 11*3 doesn't count as no 5 is involved. - Jon Perry, Jul 03 2004

Conjecture: a(n) = Floor(2*(n + 3)/3) - Floor(3*(n + 3)/5). - John W. Layman, Sep 23 2009

Also, it appears that a(n) gives the number of distinct multisets of n-1 integers, each of which is -2, +3, or +4, such that the sum of the members of each multiset is 2. E.g., for n=5, the multiset {-2,-2,3,3}, and no others, of n-1=4 members, sums to 2, so a(5)=1. - John W. Layman, Sep 23 2009

Appears to be the number of ordered triples summing to n such that 2x = 3y + 4z, ranked by A357489. An unordered version appears to be A357849, ranked by A358102. - Gus Wiseman, Nov 04 2022

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 217
• Index entries for linear recurrences with constant coefficients, signature (0,0,1,0,1,0,0,-1).

Crossrefs

Cf. A103221.

Programs

GAP

a:=[1,0,0,1,0,1,1,0];; for n in [9..100] do a[n]:=a[n-3]+a[n-5]-a[n-8]; od; a; # G. C. Greubel, Sep 08 2019

Magma

R:=PowerSeriesRing(Integers(), 100); Coefficients(R!( 1/((1-x^3)*(1-x^5)) )); // G. C. Greubel, Sep 08 2019

Maple

a := proc (n) option remember; if n < 0 then return 0 elif n = 0 then return 1 else return a(n-3)+a(n-5)-a(n-8) end if end proc

Mathematica

CoefficientList[Series[1/((1-x^3)(1-x^5)), {x, 0, 100}], x] (* Vincenzo Librandi, Jun 23 2013 *)

PARI

Vec(O(x^99)+1/(1-x^3)/(1-x^5)) \\ Charles R Greathouse IV, Jun 20 2011

Sage

def A008676_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(1/((1-x^3)*(1-x^5))).list()
A008676_list(100) # G. C. Greubel, Sep 08 2019

Formula

G.f.: 1/( (1-x^3) * (1-x^5) ).

a(n) = a(n-3) + a(n-5) - a(n-8), a(0)=a(3)=a(5)=a(6)=1, a(1)=a(2)=a(4) =a(6)=a(7)=0.

a(n) = floor((2*n+5)/5) - floor((n+2)/3). - Tani Akinari, Aug 07 2013

Extensions

Edited by Andrew Baxter, Jun 20 2011

Typo in name fixed by Vincenzo Librandi, Jun 23 2013