A008680 Expansion of 1/((1-x^3)*(1-x^4)*(1-x^5)).

1, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 7, 8, 8, 8, 9, 10, 10, 11, 11, 12, 13, 13, 14, 15, 15, 16, 17, 18, 18, 19, 20, 21, 22, 22, 23, 25, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 37, 38, 40, 41, 42, 43, 44, 46, 47, 48, 49, 51, 52, 53, 55, 56, 57

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Number of partitions of n into parts 3, 4, and 5. - Joerg Arndt, Aug 17 2013

Examples

			G.f. = 1 + x^3 + x^4 + x^5 + x^6 + x^7 + 2*x^8 + 2*x^9 + 2*x^10 + 2*x^11 + ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vincenzo Librandi)
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 228
• Index entries for linear recurrences with constant coefficients, signature (0,0,1,1,1,0,-1,-1,-1,0,0,1).

Programs

GAP

a:=[1,0,0,1,1,1,1,1,2,2,2,2];; for n in [13..80] do a[n]:=a[n-3] +a[n-4]+a[n-5]-a[n-7]-a[n-8]-a[n-9]+a[n-12]; od; a; # G. C. Greubel, Sep 09 2019

Magma

R:=PowerSeriesRing(Integers(), 80); Coefficients(R!( 1/((1-x^3)*(1-x^4)*(1-x^5)) )); // G. C. Greubel, Sep 09 2019

Maple

a:= proc(n) local m, r; m:= iquo(n, 60, 'r'); r:= r+1; (5+r+30*m)*m+ [1, 0$2, 1$5, 2$4, 3$3, 4$3, 5$2, 6$3, 7, 8$3, 9, 10$2, 11$2, 12, 13$2, 14, 15$2, 16, 17, 18$2, 19, 20, 21, 22$2, 23, 25, i$i=25..35][r] end: seq(a(n), n=0..100);  # Alois P. Heinz, Oct 06 2008

Mathematica

CoefficientList[Series[1/((1-x^3)(1-x^4)(1-x^5)),{x,0,80}],x] (* Harvey P. Dale, Apr 29 2011 *)

PARI

{a(n) = if( n<0, n=-12-n); polcoeff( 1 / ((1 - x^3) * (1 - x^4) * (1 - x^5)) + x * O(x^n), n)}; /* Michael Somos, Aug 13 2007 */

Sage

def A008680_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(1/((1-x^3)*(1-x^4)*(1-x^5))).list()
A008680_list(80) # G. C. Greubel, Sep 09 2019

Formula

Euler transform of length 5 sequence [ 0, 0, 1, 1, 1]. - Michael Somos, Aug 13 2007

From Michael Somos, Aug 13 2007: (Start)

G.f.: 1 / ((1 - x^3) * (1 - x^4) * (1 - x^5)).

a(n) = a(-12-n) for all n in Z. (End)

a(n) = floor((1+(-1)^n)*(-1)^floor(n/2)/8 +(n^2+12*n+90)/120). - Tani Akinari, Aug 17 2013

Extensions

Typo in name fixed by Vincenzo Librandi, Jun 23 2013