G. C. Greubel, <a href="/A192421/b192421_1.txt">Table of n, a(n) for n = 0..1000</a>

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From these, read A192421 a(n) = (2, 0, 3, 1, 8, ...) and A192422 = (0, 1, 1, 5, 7, ...).

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

The polynomial p(n,x) is defined by ((x+d)/2)^n + ((x-d)/2)^n, where d = sqrt(x^2+4). For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232.

G. C. Greubel, <a href="/A192421/b192421_1.txt">Table of n, a(n) for n = 0..1000</a>

<a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-1,-1).

From Colin Barker, May 12 2014: (Start)

Conjecture: a(n) = a(n-1) + 3*a(n-2) - a(n-3) - a(n-4). G.f.: -(3*x^2+2*x-2) / (x^4+x^3-3*x^2-x+1). - _Colin Barker_, May 12 2014

G.f.: (2-2*x-3*x^2)/(1-x-3*x^2+x^3+x^4). (End)

a(n) = Sum_{j=0..n} T(n, j)*Fibonacci(j-1), where T(n, k) = [x^k] ((x + sqrt(x^2+4))^n + (x - sqrt(x^2+4))^n)/2^n. - G. C. Greubel, Jul 11 2023

p(0,x) = 2 -> 2.

p(1,x) = x -> x.

p(2,x) = 2 + x^2 -> 3 + x.

p(3,x) =3x 3*x + x^3 -> 1 +5x 5*x.

p(4,x) = 2 +4x 4*x^2 + x^4 -> 8 +7x 7*x.

From these, read A192421 = (2, 0, 3, 1, 8, ...) and A192422 = (0, 1, 1, 5, 7, ...).

q[x_] := x + 1; d = Sqrt[x^2 + 4];

p[n_, x_] := ((x + d)/2)^n + ((x - d)/2)^n (* A162514 *)

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

Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}] (* A192421 *)

Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}] (* A192422 *)

LinearRecurrence[{1, 3, -1, -1}, {2, 0, 3, 1}, 40] (* G. C. Greubel, Jul 11 2023 *)

(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (2-2*x-3*x^2)/(1-x-3*x^2+x^3+x^4) )); // G. C. Greubel, Jul 11 2023

(SageMath)

@CachedFunction

def a(n): # a = A192421

if (n<4): return (2, 0, 3, 1)[n]

else: return a(n-1) +3*a(n-2) -a(n-3) -a(n-4)

[a(n) for n in range(41)] # G. C. Greubel, Jul 11 2023

Cf. A162514, A192232, A192422.

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p[n_, x_] := ((x + d)/2)^n + ((x - d)/2)^n (* A161514 A162514 *)

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