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a(n) = (a(n-1)+1)*(a(n-3)+1)/a(n-4) for n > 3, a(0) = a(1) = a(2) = a(3) = 1.
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%I #35 Apr 24 2021 03:31:39

%S 1,1,1,1,4,10,22,115,319,736,3886,10816,24991,131989,367405,848947,

%T 4483720,12480934,28839196,152314471,423984331,979683706,5174208274,

%U 14402986300,33280406797,175770766825,489277549849,1130554147381,5971031863756,16621033708546

%N a(n) = (a(n-1)+1)*(a(n-3)+1)/a(n-4) for n > 3, a(0) = a(1) = a(2) = a(3) = 1.

%H Colin Barker, <a href="/A276308/b276308.txt">Table of n, a(n) for n = 0..1000</a>

%H S. Fomin and A. Zelevinsky, <a href="http://dx.doi.org/10.1006/aama.2001.0770">The Laurent Phenomenon</a>, Advances in Applied Mathematics, 28 (2002), 119-144.

%H Matthew Christopher Russell, <a href="https://www.semanticscholar.org/paper/Using-experimental-mathematics-to-conjecture-and-in-Russell/fdebe20954dacb7ec7a24afe2cf491b951c5a28d">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 Eq. (6.137).

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,35,0,0,-35,0,0,1).

%F From _Colin Barker_, Aug 29 2016: (Start)

%F a(n) = 35*a(n-3)-35*a(n-6)+a(n-9) for n>8.

%F G.f.: (1+x+x^2-34*x^3-31*x^4-25*x^5+22*x^6+10*x^7+4*x^8) / ((1-x)*(1+x+x^2)*(1-34*x^3+x^6)).

%F (End)

%o (Ruby)

%o def A(m, n)

%o a = Array.new(m, 1)

%o ary = [1]

%o while ary.size < n + 1

%o i = (a[1] + 1) * (a[-1] + 1)

%o break if i % a[0] > 0

%o a = *a[1..-1], i / a[0]

%o ary << a[0]

%o end

%o ary

%o end

%o def A276308(n)

%o A(4, n)

%o end

%o (PARI) Vec((1+x+x^2-34*x^3-31*x^4-25*x^5+22*x^6+10*x^7+4*x^8)/((1-x)*(1+x+x^2)*(1-34*x^3+x^6)) + O(x^35)) \\ _Colin Barker_, Aug 29 2016

%o (PARI) a276308(maxn) = {a=vector(maxn); a[1]=a[2]=a[3]=a[4]=1; for(n=5, maxn, a[n]=(a[n-1]+1)*(a[n-3]+1)/a[n-4]); a} \\ _Colin Barker_, Aug 30 2016

%Y Cf. A276123, A276175.

%K nonn,easy

%O 0,5

%A _Seiichi Manyama_, Aug 29 2016