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A192386 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments. 3

%I #19 Jul 10 2023 08:21:09

%S 1,0,8,8,96,224,1408,4608,22784,86016,386048,1548288,6676480,27467776,

%T 116490240,484409344,2040135680,8521777152,35786063872,149761818624,

%U 628140015616,2630784909312,11028578435072,46205266558976,193656954814464

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

%C The polynomial p(n,x) is defined by ((x+d)^n - (x-d)^n)/(2*d), where d = sqrt(x+5). For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232.

%H G. C. Greubel, <a href="/A192386/b192386.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,12,-8,-16).

%F From _Colin Barker_, May 11 2014: (Start)

%F a(n) = 2*a(n-1) + 12*a(n-2) - 8*a(n-3) - 16*a(n-4).

%F G.f.: x*(1-2*x-4*x^2)/(1-2*x-12*x^2+8*x^3+16*x^4). (End)

%F From _G. C. Greubel_, Jul 10 2023: (Start)

%F T(n, k) = [x^k] ((x+sqrt(x+5))^n - (x-sqrt(x+5))^n)/(2*sqrt(x+5)).

%F a(n) = Sum_{k=0..n-1} T(n, k)*Fibonacci(k-1). (End)

%e The first five polynomials p(n,x) and their reductions are as follows:

%e p(0, x) = 1 -> 1

%e p(1, x) = 2*x -> 2*x

%e p(2, x) = 3 + x + 3*x^2 -> 8 + 4*x

%e p(3, x) = 12*x + 4*x^2 + 4*x^3 -> 8 + 32*x

%e p(4, x) = 9 + 6*x + 31*x^2 + 10*x^3 + 5*x^4 -> 96 + 96*x.

%e From these, read A192386 = (1, 0, 8, 8, 96, ...) and A192387 = (0, 2, 4, 32, 96, ...).

%t q[x_]:= x+1; d= Sqrt[x+5];

%t p[n_, x_]:= ((x+d)^n - (x-d)^n)/(2*d); (* suggested by A162517 *)

%t Table[Expand[p[n, x]], {n,6}]

%t reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)};

%t t = Table[ FixedPoint[Expand[#1 /. reductionRules] &, p[n, x]], {n, 1, 30}];

%t Table[Coefficient[Part[t, n], x, 0], {n,30}] (* A192386 *)

%t Table[Coefficient[Part[t, n], x, 1], {n,30}] (* A192387 *)

%t Table[Coefficient[Part[t, n]/2, x, 1], {n,30}] (* A192388 *)

%t LinearRecurrence[{2,12,-8,-16}, {1,0,8,8}, 40] (* _G. C. Greubel_, Jul 10 2023 *)

%o (Magma)

%o R<x>:=PowerSeriesRing(Integers(), 41);

%o Coefficients(R!( x*(1-2*x-4*x^2)/(1-2*x-12*x^2+8*x^3+16*x^4) )); // _G. C. Greubel_, Jul 10 2023

%o (SageMath)

%o @CachedFunction

%o def a(n): # a = A192386

%o if (n<5): return (0,1,0,8,8)[n]

%o else: return 2*a(n-1) +12*a(n-2) -8*a(n-3) -16*a(n-4)

%o [a(n) for n in range(1,41)] # _G. C. Greubel_, Jul 10 2023

%Y Cf. A083087, A192232, A192387, A192388.

%K nonn,easy

%O 1,3

%A _Clark Kimberling_, Jun 30 2011

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Last modified March 28 08:12 EDT 2024. Contains 371236 sequences. (Running on oeis4.)