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%I #29 Jan 06 2024 18:33:16
%S 1,1,1,1,1,1,2,3,4,6,9,22,36,51,82,129,321,529,753,1217,1921,4786,
%T 7891,11236,18166,28681,71462,117828,167779,271266,428289,1067137,
%U 1759521,2505441,4050817,6395649,15935586,26274979,37413828,60490982,95506441,237966646
%N a(n) = (a(n-3) + a(n-1) * a(n-5)) / a(n-6), a(0) = a(1) = ... = a(5) = 1.
%C Inspired by A048736.
%H Colin Barker, <a href="/A275173/b275173.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,16,0,0,0,0,-16,0,0,0,0,1).
%F G.f.: (1 +x +x^2 +x^3 +x^4 -15*x^5 -14*x^6 -13*x^7 -12*x^8 -10*x^9 +9*x^10 +6*x^11 +4*x^12 +3*x^13 +2*x^14) / ((1 -x)*(1 +x +x^2 +x^3 +x^4)*(1 -15*x^5 +x^10)). - _Colin Barker_, Jul 19 2016
%F a(n) = 16*a(n-5) - 16*a(n-10) + a(n-15). - _G. C. Greubel_, Jul 20 2016
%t RecurrenceTable[{a[n] == (a[n - 3] + a[n - 1] a[n - 5])/a[n - 6], a[1] == 1, a[2] == 1, a[3] == 1, a[4] == 1, a[5] == 1, a[6] == 1}, a, {n, 42}] (* or *)
%t CoefficientList[Series[(1 + x + x^2 + x^3 + x^4 - 15 x^5 - 14 x^6 - 13 x^7 - 12 x^8 - 10 x^9 + 9 x^10 + 6 x^11 + 4 x^12 + 3 x^13 + 2 x^14)/((1 - x) (1 + x + x^2 + x^3 + x^4) (1 - 15 x^5 + x^10)), {x, 0, 41}], x] (* _Michael De Vlieger_, Jul 19 2016 *)
%t nxt[{a_,b_,c_,d_,e_,f_}]:={b,c,d,e,f,(d+f*b)/a}; NestList[nxt,{1,1,1,1,1,1},50][[;;,1]] (* _Harvey P. Dale_, Jan 06 2024 *)
%o (Ruby)
%o def A(k, l, n)
%o a = Array.new(k * 2, 1)
%o ary = [1]
%o while ary.size < n + 1
%o break if (a[1] * a[-1] + a[k] * l) % a[0] > 0
%o a = *a[1..-1], (a[1] * a[-1] + a[k] * l) / a[0]
%o ary << a[0]
%o end
%o ary
%o end
%o def A275173(n)
%o A(3, 1, n)
%o end
%o (PARI) Vec((1 +x +x^2 +x^3 +x^4 -15*x^5 -14*x^6 -13*x^7 -12*x^8 -10*x^9 +9*x^10 +6*x^11 +4*x^12 +3*x^13 +2*x^14) / ((1 -x)*(1 +x +x^2 +x^3 +x^4)*(1 -15*x^5 +x^10)) + O(x^50)) \\ _Colin Barker_, Jul 19 2016
%Y Cf. A101879, A048736, A275174.
%Y Variants: A275175, A275176.
%K nonn,easy
%O 0,7
%A _Seiichi Manyama_, Jul 19 2016
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