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A025795 Expansion of 1/((1-x^2)(1-x^3)(1-x^5)). 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

a(n) = 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

Table of n, a(n) for n=0..74.

M. Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, 2014; http://matinf.pmfbl.org/wp-content/uploads/2015/01/za-arhiv-18.-1.pdf

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 have also : 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*5^0.5+25))*cos((2*n*Pi)/5)-(4/(5*5^0.5-25))*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(-10-n) = a(n). - Michael Somos, Feb 25 2008

EXAMPLE

1 + q^2 + q^3 + q^4 + 2*q^5 + 2*q^6 + 2*q^7 + 3*q^8 + 3*q^9 + 4*q^10 + ...

PROG

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

CROSSREFS

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

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Last modified December 10 04:09 EST 2016. Contains 278993 sequences.