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%I A192426 #21 Jul 14 2023 09:04:46
%S A192426 2,0,5,1,18,13,81,106,413,729,2258,4653,12833,28666,74493,173545,
%T A192426 437346,1041421,2583089,6221322,15304541,37079289,90826994,220729069,
%U A192426 539487297,1313161498,3205831869,7809748489,19054635650,46439068365
%N A192426 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments.
%C A192426 The polynomial p(n,x) is defined by ((x+d)/2)^n + ((x-d)/2)^n, where d = sqrt(x^2+8). For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232.
%H A192426 G. C. Greubel, <a href="/A192426/b192426.txt">Table of n, a(n) for n = 0..1000</a>
%H A192426 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,5,-2,-4).
%F A192426 From _Colin Barker_, May 12 2014: (Start)
%F A192426 a(n) = a(n-1) + 5*a(n-2) - 2*a(n-3) - 4*a(n-4).
%F A192426 G.f.: (2-2*x-5*x^2)/(1-x-5*x^2+2*x^3+4*x^4). (End)
%F A192426 a(n) = Sum_{k=0..n} T(n, k)*Fibonacci(k-1), where T(n, k) = [x^k] ((x + sqrt(x^2+8))^n + (x - sqrt(x^2+8))^n)/2^n. - _G. C. Greubel_, Jul 12 2023
%e A192426 The first five polynomials p(n,x) and their reductions are as follows:
%e A192426   p(0,x) = 2 -> 2
%e A192426   p(1,x) = x -> x
%e A192426   p(2,x) = 4 + x^2 -> 5 + x
%e A192426   p(3,x) = 6*x + x^3 -> 1 + 8*x
%e A192426   p(4,x) = 8 + 8*x^2 + x^4 -> 18 + 11*x.
%e A192426 From these, read a(n) = (2, 0, 5, 1, 18, ...) and A192427 = (0, 1, 1, 8, 11, ...).
%t A192426 q[x_]:= x+1; d= Sqrt[x^2+8];
%t A192426 p[n_, x_]:= ((x+d)/2)^n + ((x-d)/2)^n (* suggested by A162514 *)
%t A192426 Table[Expand[p[n, x]], {n, 0, 6}]
%t A192426 reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)};
%t A192426 t= Table[FixedPoint[Expand[#1/. reductionRules] &, p[n,x]], {n,0,30}]
%t A192426 Table[Coefficient[Part[t, n], x, 0], {n,30}] (* A192426 *)
%t A192426 Table[Coefficient[Part[t, n], x, 1], {n,30}] (* A192427 *)
%t A192426 LinearRecurrence[{1,5,-2,-4}, {2,0,5,1}, 40] (* _G. C. Greubel_, Jul 12 2023 *)
%o A192426 (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (2-2*x-5*x^2)/(1-x-5*x^2+2*x^3+4*x^4) )); // _G. C. Greubel_, Jul 12 2023
%o A192426 (SageMath)
%o A192426 @CachedFunction
%o A192426 def a(n): # a = A192426
%o A192426     if (n<4): return (2,0,5,1)[n]
%o A192426     else: return a(n-1) + 5*a(n-2) - 2*a(n-3) - 4*a(n-4)
%o A192426 [a(n) for n in range(41)] # _G. C. Greubel_, Jul 12 2023
%Y A192426 Cf. A000045, A162514, A192232, A192427.
%K A192426 nonn,easy
%O A192426 0,1
%A A192426 _Clark Kimberling_, Jun 30 2011
%E A192426 Typo in name corrected by _G. C. Greubel_, Jul 12 2023

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