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

%I A025795 #54 Jun 16 2025 10:11:42
%S A025795 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 A025795 19,21,21,23,24,25,27,28,29,31,32,34,35,37,38,40,42,43,45,47,48,51,52,
%U A025795 54,56,58,60,62,64,66,68,71,72,75,77,79,82,84,86,89,91,94,96,99,101,104
%N A025795 Expansion of 1/((1-x^2)*(1-x^3)*(1-x^5)).
%C A025795 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
%C A025795 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
%H A025795 Alois P. Heinz, <a href="/A025795/b025795.txt">Table of n, a(n) for n = 0..10000</a>
%H A025795 Milan 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.
%H A025795 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,1,0,0,0,-1,-1,0,1).
%F A025795 G.f.: 1/((1-x^2)*(1-x^3)*(1-x^5)).
%F A025795 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 A025795 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
%F A025795 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
%F A025795 Euler transform of length 5 sequence [0, 1, 1, 0, 1]. - _Michael Somos_, Feb 05 2008
%F A025795 a(n) = a(-10-n) for all n in Z. - _Michael Somos_, Feb 25 2008
%F A025795 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
%F A025795 a(n)-a(n-3) = A008616(n). - _R. J. Mathar_, Jun 23 2021
%F A025795 a(n) = floor((n^2 + 10*n + 6*(9+(-1)^n))/60). - _Hoang Xuan Thanh_, Jun 15 2025
%e A025795 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 A025795 a[ n_] := Quotient[n^2 + 10 n + 1 - 13 Mod[n, 2], 60] + 1; (* _Michael Somos_, Nov 17 2017 *)
%o A025795 (PARI) {a(n) = (n^2 + 10*n + 1 - n%2 * 13) \60 + 1} /* _Michael Somos_, Feb 05 2008 */
%Y A025795 Cf. A008676, A078495, A103221.
%K A025795 nonn,easy
%O A025795 0,6
%A A025795 _N. J. A. Sloane_