A025795 Expansion of 1/((1-x^2)*(1-x^3)*(1-x^5)).

1, 0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 5, 5, 6, 7, 7, 8, 9, 9, 11, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 21, 23, 24, 25, 27, 28, 29, 31, 32, 34, 35, 37, 38, 40, 42, 43, 45, 47, 48, 51, 52, 54, 56, 58, 60, 62, 64, 66, 68, 71, 72, 75, 77, 79, 82, 84, 86, 89, 91, 94, 96, 99, 101, 104

Offset

0,6

Comments

a(n) is the number of ways to pay n dollars with coins of two, three and five dollars. E.g., a(0)=1 because there is one way to pay: with no coin; a(1)=0 no possibility; a(2)=1 (2=1*2); a(3)=1 (3=1*3); a(4)=1 (4=2*2) a(5)=2 (5=3+2=1*5) ... - Richard Choulet, Jan 20 2008

a(n) is the number of partitions of n into parts which are 2, 3, or 5 (inclusive or). a(0)=1 by definition. See the preceding comment by R. Choulet. - Wolfdieter Lang, Mar 15 2012

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

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 (0,1,1,0,0,0,-1,-1,0,1).

Formula

G.f.: 1/((1-x^2)(1-x^3)(1-x^5)).

Let [b(1); b(2); ...; b(p)] denote a periodic sequence: e.g., [0; 1] defines the sequence c such that c(0)=c(2)=...=c(2*k)=0 and c(1)=c(3)=...=c(2*k+1)=1. Then a(n)=0.25*[0; 1] - (1/3)*[1; 0; 0] + (1/5)*[0; 1; 1; 0; 3] + ((n+1)*(n+2)/60) + (7*(n+1)/60). - Richard Choulet, Jan 20 2008

If ||A|| is the nearest number to A (A not a half-integer) we also have a(n) = ||((n+1)*(n+9)/60) + (1/5)[0; 1; 1; 0; 3]. - Richard Choulet, Jan 20 2008

a(n) = 77/360 + 7*(n+1)/60 + (n+2)*(n+1)/60 + (-1)^n/8 - (2/9)*cos(2*(n+2)*Pi/3) + (4/(5*sqrt(5)+25))*cos(2*n*Pi/5) - (4/(5*sqrt(5)-25))*cos(4*n*Pi/5). - Richard Choulet, Jan 20 2008

Euler transform of length 5 sequence [0, 1, 1, 0, 1]. - Michael Somos, Feb 05 2008

a(n) = a(-10-n) for all n in Z. - Michael Somos, Feb 25 2008

a(n) - a(n-2) = A008676(n). a(n) - a(n-5) = A103221(n) = A008615(n+2). A078495(n) = 2^(a(n-7) + a(n-9)) * 3^a(n-8) for all n in Z. - Michael Somos, Nov 17 2017, corrected Jun 23 2021

a(n)-a(n-3) = A008616(n). - R. J. Mathar, Jun 23 2021

Example

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

Mathematica

a[ n_] := Quotient[n^2 + 10 n + 1 - 13 Mod[n, 2], 60] + 1; (* Michael Somos, Nov 17 2017 *)

Prog

(PARI) {a(n) = (n^2 + 10*n + 1 - n%2 * 13) \60 + 1} /* Michael Somos, Feb 05 2008 */

Crossrefs

Cf. A008676, A078495, A103221.

Sequence in context: A078452 A263997 A135636 * A219610 A194161 A051066

Adjacent sequences: A025792 A025793 A025794 * A025796 A025797 A025798

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved