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A192428 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments.

%I A192428 #12 Jul 14 2023 09:04:40
%S A192428 1,1,5,11,57,185,829,3067,12801,49633,201413,794747,3190617,12673529,
%T A192428 50672029,201782923,805529409,3210794113,12810136517,51078991403,
%U A192428 203744818617,812521585145,3240726179389,12924488375899,51547405667265
%N A192428 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments.
%C A192428 The polynomial p(n,x) is defined by ((x+d)^n + (x-d)^n)/2 + ((x+d)^n - (x-d)^n)/(2*d), where d = sqrt(x^2+4), as at A163762. For an introduction to reductions of polynomials by substitutions such as x^2 -> x+2, see A192232.
%H A192428 G. C. Greubel, <a href="/A192428/b192428.txt">Table of n, a(n) for n = 0..1000</a>
%H A192428 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,10,-6,-9).
%F A192428 From _Colin Barker_, May 12 2014: (Start)
%F A192428 a(n) = 2*a(n-1) + 10*a(n-2) - 6*a(n-3) - 9*a(n-4).
%F A192428 G.f.: (1-x-7*x^2-3*x^3)/(1-2*x-10*x^2+6*x^3+9*x^4). (End)
%F A192428 a(n) = Sum_{k=0..n} T(n,k)*Fibonacci(k-1), where T(n, k) = [x^k] ( ((x + sqrt(x+4))^n + (x - sqrt(x+4))^n)/2 + ((x + sqrt(x+4))^n - (x - sqrt(x+4))^n)/(2*sqrt(x+4)) ). - _G. C. Greubel_, Jul 13 2023
%e A192428 The first five polynomials p(n,x) and their reductions are as follows:
%e A192428   p(0,x) = 1 -> 1
%e A192428   p(1,x) = 1 + x -> 1 + x
%e A192428   p(2,x) = 4 + 3*x + x^2 -> 5 + 4*x
%e A192428   p(3,x) = 4 + 13*x + 6*x^2 + x^3 -> 11 + 21*x
%e A192428   p(4,x) = 16 + 24*x + 29*x^2 + 10*x^3 + x^4 -> 57 + 76*x.
%e A192428 From these, read a(n) = (1, 1, 5, 11, 57, 185, ...) and A192429 = (0, 1, 4, 21, 76, 329, ...).
%t A192428 q[x_]:= x+1; d= Sqrt[x+4];
%t A192428 u[x_]:= x+d; v[x_]:= x-d;
%t A192428 p[n_, x_]:= (u[x]^n +v[x]^n)/2 + (u[x]^n -v[x]^n)/(2*d) (* A163762 *)
%t A192428 Table[Expand[p[n, x]], {n, 0, 6}]
%t A192428 reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)};
%t A192428 t = Table[FixedPoint[Expand[#1 /. reductionRules] &, p[n, x]], {n, 0, 30}]
%t A192428 Table[Coefficient[Part[t, n], x, 0], {n,30}] (* A192428 *)
%t A192428 Table[Coefficient[Part[t, n], x, 1], {n,30}] (* A192429 *)
%t A192428 LinearRecurrence[{2,10,-6,-9}, {1,1,5,11}, 40] (* _G. C. Greubel_, Jul 13 2023 *)
%o A192428 (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x-7*x^2-3*x^3)/(1-2*x-10*x^2+6*x^3+9*x^4) )); // _G. C. Greubel_, Jul 13 2023
%o A192428 (SageMath)
%o A192428 @CachedFunction
%o A192428 def a(n): # a = A192428
%o A192428     if (n<4): return (1, 1, 5, 11)[n]
%o A192428     else: return 2*a(n-1) + 10*a(n-2) - 6*a(n-3) - 9*a(n-4)
%o A192428 [a(n) for n in range(41)] # _G. C. Greubel_, Jul 13 2023
%Y A192428 Cf. A163762, A192232, A192429.
%K A192428 nonn
%O A192428 0,3
%A A192428 _Clark Kimberling_, Jun 30 2011