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%I #20 Dec 22 2023 09:41:05
%S 0,0,0,0,0,0,0,0,2,0,0,3,0,3,4,0,6,5,4,10,6,10,15,12,20,21,23,35,34,
%T 44,56,57,80,91,101,137,148,181,230,249,318,379,430,549,629,748,928,
%U 1060,1298,1557,1809,2226,2617,3109,3783,4426,5336,6400,7536,9120
%N Number of compositions of n into parts 3 and 5 with at least one 3 and one 5.
%H Reinhard Zumkeller, <a href="/A245492/b245492.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (-1, -1, 1, 1, 3, 2, 2, -1, -1, -2, -1, -1).
%F a(n) = a(n-3)+a(n-5)+b(n) where b(n) is the 15-cycle: (1,1,0,0,1,1,0,1,0,0,2,0,0,1,0) with b(n)=b(n-15) starting at b(13)=1. e.g. b(28)=b(13). The initial values for a(n) are: a(8)=2, a(9)=0, a(10)=0, a(11)=3, a(12)=0.
%F G.f.: x^8*(x^4+x^3+2*x^2+2*x+2) / ((x-1)*(x^2+x+1)*(x^4+x^3+x^2+x+1)*(x^5+x^3-1)). - _Colin Barker_, Jul 24 2014
%e a(20)=6, the tuples being: (533333),(353333),(335333),(333533),(333353),(333335).
%t CoefficientList[Series[x^8*(x^4 + x^3 + 2*x^2 + 2*x + 2)/((x - 1)*(x^2 + x + 1)*(x^4 + x^3 + x^2 + x + 1)*(x^5 + x^3 - 1)), {x, 0, 60}], x] (* _Wesley Ivan Hurt_, Jul 24 2014 *)
%o (PARI) a=[0,0,0,0,0,0,0,2,0,0,3,0]; b=[1,1,0,0,1,1,0,1,0,0,2,0,0,1,0]; k=1; for(n=13, 100, a=concat(a, a[n-3]+a[n-5]+b[k]); if(k==#b, k=1, k++)); a \\ _Colin Barker_, Jul 24 2014
%o (Haskell)
%o a245492 n = a245492_list !! (n-1)
%o a245492_list = [0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 3, 0] ++
%o zipWith3 (((+) .) . (+))
%o (drop 8 a245492_list) (drop 10 a245492_list)
%o (cycle [1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 2, 0, 0, 1, 0])
%o -- _Reinhard Zumkeller_, Jul 28 2014
%Y Cf. A245332, A245487.
%K nonn,easy
%O 0,9
%A _David Neil McGrath_, Jul 24 2014
%E More terms from _Colin Barker_, Jul 24 2014
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