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%I #39 Nov 06 2020 10:38:01
%S 1,1,1,1,1,6,16,41,106,806,2311,6126,16066,122401,351136,931006,
%T 2441881,18604041,53370241,141506681,371149801,2827691726,8111925376,
%U 21508084401,56412327826,429790538206,1232959286791,3269087322166,8574302679706,65325334115481
%N a(n) = (1 + Sum_{j=1..K-1} a(n-j) + a(n-1)*a(n-K+1))/a(n-K) with a(1),...,a(K)=1, where K=5.
%H Seiichi Manyama, <a href="/A283330/b283330.txt">Table of n, a(n) for n = 1..1838</a>
%H Matthew Christopher Russell, <a href="http://www.math.rutgers.edu/~zeilberg/Theses/MatthewRussellThesis.pdf">Using experimental mathematics to conjecture and prove theorems in the theory of partitions and commutative and non-commutative recurrences</a>, PhD Dissertation, Mathematics Department, Rutgers University, May 2016; see <a href="https://pdfs.semanticscholar.org/fdeb/e20954dacb7ec7a24afe2cf491b951c5a28d.pdf">also</a>.
%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,153,0,0,0,-153,0,0,0,1).
%F From _Seiichi Manyama_, Mar 18 2017: (Start)
%F a(4*n-1) = 3*a(4*n-2) - a(4*n-3) - 1,
%F a(4*n) = 3*a(4*n-1) - a(4*n-2) - 1,
%F a(4*n+1) = 3*a(4*n) - a(4*n-1) - 1,
%F a(4*n+2) = 8*a(4*n+1) - a(4*n) - 1. (End)
%F From _Colin Barker_, Nov 03 2020: (Start)
%F G.f.: x*(1 + x + x^2 + x^3 - 152*x^4 - 147*x^5 - 137*x^6 - 112*x^7 + 106*x^8 + 41*x^9 + 16*x^10 + 6*x^11) / ((1 - x)*(1 + x)*(1 + x^2)*(1 - 152*x^4 + x^8)).
%F a(n) = 153*a(n-4) - 153*a(n-8) + a(n-12) for n>12.
%F (End)
%t a[n_] := a[n] = If[n <= 5, 1, With[{m = If[Mod[n, 4] == 2, 8, 3]}, m a[n-1] - a[n-2] - 1]];
%t Array[a, 30] (* _Jean-François Alcover_, Nov 03 2020 *)
%o (Ruby)
%o def A(k, n)
%o a = Array.new(k, 1)
%o ary = [1]
%o while ary.size < n
%o j = (1..k - 1).inject(1){|s, i| s + a[-i]} + a[1] * a[-1]
%o break if j % a[0] > 0
%o a = *a[1..-1], j / a[0]
%o ary << a[0]
%o end
%o ary
%o end
%o def A283330(n)
%o A(5, n)
%o end # _Seiichi Manyama_, Mar 18 2017
%Y Cf. A276123, A283329.
%K nonn
%O 1,6
%A _N. J. A. Sloane_, Mar 17 2017
%E More terms from _Seiichi Manyama_, Mar 17 2017
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