A008677 - OEIS

A008677

Expansion of 1/((1-x^3)*(1-x^5)*(1-x^7)).

1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 3, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 9, 9, 9, 10, 10, 11, 11, 12, 12, 12, 14, 13, 14, 15, 15, 16, 16, 17, 17, 18, 19, 19, 20, 20, 21, 22, 22, 23, 24, 24, 25, 26, 26, 27, 28, 29, 29, 30, 31, 31, 33, 33, 34, 35, 35, 37, 37, 38, 39, 40, 41, 41, 43

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OFFSET

0,11

COMMENTS

Number of partitions of n into parts 3, 5, and 7. - Joerg Arndt, Aug 17 2013

Number of different total numbers of kicks, tries and converted tries which lead to a score of n in a rugby (union) match. - Matthew Scroggs, Jul 09 2015

REFERENCES

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 114, [6t].

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 230

M. Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.

Index entries for linear recurrences with constant coefficients, signature (0,0,1,0,1,0,1,-1,0,-1,0,-1,0,0,1).

FORMULA

a(n) = a(n-3) + a(n-5) + a(n-7) - a(n-8) - a(n-10) - a(n-12) + a(n-15) for n >= 15. - David Neil McGrath, Sep 03 2014

G.f.: 1 / ((1 - x^3) * (1 - x^5) * (1 - x^7)).

Euler transform of length 7 sequence [ 0, 0, 1, 0, 1, 0, 1]. - Michael Somos, Sep 30 2014

a(n) = a(-15-n) for all n in Z. - Michael Somos, Sep 30 2014

0 = a(n) - a(n+3) - a(n+5) + a(n+8) - [mod(n, 7) == 6] for all n in Z. - Michael Somos, Sep 30 2014

a(n) = round(n^2/210 + n/14 + 5/21) + r(n) where r(n) = 1 if n == 0, 3, 10, 15, 45, 75, 80, 87, or 90 (mod 105), r(n) = -1 if n == 4, 11, 16, 44, 46, 74, 79 or 86 (mod 105), r(n) = 0 otherwise. - Robert Israel, Jul 09 2015

EXAMPLE

G.f. = 1 + x^3 + x^5 + x^6 + x^7 + x^8 + x^9 + 2*x^10 + x^11 + 2*x^12 + ...

MAPLE

S:= series(1/((1-x^3)*(1-x^5)*(1-x^7)), x, 101):

seq(coeff(S, x, j), j=0..100); # Robert Israel, Jul 09 2015

MATHEMATICA

CoefficientList[Series[1/((1-x^3)(1-x^5)(1-x^7)), {x, 0, 100}], x] (* Vincenzo Librandi, Jun 23 2013 *)

LinearRecurrence[{0, 0, 1, 0, 1, 0, 1, -1, 0, -1, 0, -1, 0, 0, 1}, {1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2}, 70] (* Harvey P. Dale, Aug 04 2020 *)

PROG

(PARI) a(n)=[1, 0, -2, 2, -2, 0, 1][n%7+1]/7 +[2, -1, 0, 0, -1][n%5+1]/5 +[2, -1, -1][n%3+1]/9 +(3*n^2+45*n+148)/630; \\ Tani Akinari, Aug 17 2013

(PARI) a(n)=floor((n^2+15*n+86)/210+(n%3<1)/3+3*(n%5<1)/5) \\ Tani Akinari, Sep 30 2014

(Magma) R<x>:=PowerSeriesRing(Integers(), 100); Coefficients(R!( 1/((1-x^3)*(1-x^5)*(1-x^7)) )); // G. C. Greubel, Sep 08 2019

(Sage)

def A008677_list(prec):

P.<x> = PowerSeriesRing(ZZ, prec)

return P(1/((1-x^3)*(1-x^5)*(1-x^7))).list()

A008677_list(100) # G. C. Greubel, Sep 08 2019

CROSSREFS

Sequence in context: A343333 A230502 A280253 * A036497 A211976 A035460

Adjacent sequences: A008674 A008675 A008676 * A008678 A008679 A008680