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

%I

%S 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,

%T 19,21,21,23,24,25,27,28,29,31,32,34,35,37,38,40,42,43,45,47,48,51,52,

%U 54,56,58,60,62,64,66,68,71,72,75,77,79,82,84,86,89,91,94,96,99,101,104

%N Expansion of 1/((1-x^2)(1-x^3)(1-x^5)).

%C 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

%C 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

%H <a href="/index/Rec#order_10">Index to sequences with linear recurrences with constant coefficients</a>, signature (0,1,1,0,0,0,-1,-1,0,1).

%H M. Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Janjic/janjic63.html">On Linear Recurrence Equations Arising from Compositions of Positive Integers</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.

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

%F 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

%F 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

%F 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

%F Euler transform of length 5 sequence [0, 1, 1, 0, 1]. - _Michael Somos_, Feb 05 2008

%F a(n) = a(-10-n) for all n in Z. - _Michael Somos_, Feb 25 2008

%F a(n) - a(n-2) = A008686(n). a(n) - a(n-5) = A103221(n). A078495(n) = 2^(a(n-7) + a(n-9)) * 3^a(n-8) for all n in Z. - _Michael Somos_, Nov 17 2017

%e 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 + ...

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

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

%Y Cf. A008686, A078495, A103221.

%K nonn,easy,changed

%O 0,6

%A _N. J. A. Sloane_

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