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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 Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,0,0,-1,-1,0,1). M. Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7. 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(n) = a(-10-n) for all n in Z. - Michael Somos, Feb 25 2008 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 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. A008686, A078495, A103221. Sequence in context: A078452 A263997 A135636 * A219610 A194161 A051066 Adjacent sequences:  A025792 A025793 A025794 * A025796 A025797 A025798 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified March 17 14:48 EDT 2018. Contains 300565 sequences. (Running on oeis4.)