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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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 2, 3, and 5. See the preceding comment by R. Choulet. - Wolfdieter Lang, Mar 15 2012

Examples

			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 + ...
		

Links

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

Crossrefs

Cf. A008676, A078495, A103221.

Programs

Mathematica

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

PARI

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

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

a(n) = floor((n^2 + 10*n + 6*(9+(-1)^n))/60). - Hoang Xuan Thanh, Jun 15 2025