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

1, 0, 8, 8, 96, 224, 1408, 4608, 22784, 86016, 386048, 1548288, 6676480, 27467776, 116490240, 484409344, 2040135680, 8521777152, 35786063872, 149761818624, 628140015616, 2630784909312, 11028578435072, 46205266558976, 193656954814464

Offset: 1

Author

Clark Kimberling, Jun 30 2011

Keywords

• nonn
• easy

Comments

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.

Examples

			The first five polynomials p(n,x) and their reductions are as follows:
  p(0, x) = 1 -> 1
  p(1, x) =     2*x -> 2*x
  p(2, x) = 3 +   x +  3*x^2 -> 8 + 4*x
  p(3, x) =    12*x +  4*x^2 +  4*x^3 -> 8 + 32*x
  p(4, x) = 9 + 6*x + 31*x^2 + 10*x^3 + 5*x^4 -> 96 + 96*x.
From these, read A192386 = (1, 0, 8, 8, 96, ...) and A192387 = (0, 2, 4, 32, 96, ...).
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (2,12,-8,-16).

Crossrefs

Cf. A083087, A192232, A192387, A192388.

Programs

Magma

R:=PowerSeriesRing(Integers(), 41);
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

Mathematica

q[x_]:= x+1; d= Sqrt[x+5];
p[n_, x_]:= ((x+d)^n - (x-d)^n)/(2*d); (* suggested by A162517 *)
Table[Expand[p[n, x]], {n,6}]
reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[ FixedPoint[Expand[#1 /. reductionRules] &, p[n, x]], {n, 1, 30}];
Table[Coefficient[Part[t, n], x, 0], {n,30}] (* A192386 *)
Table[Coefficient[Part[t, n], x, 1], {n,30}] (* A192387 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n,30}] (* A192388 *)
LinearRecurrence[{2,12,-8,-16}, {1,0,8,8}, 40] (* G. C. Greubel, Jul 10 2023 *)

SageMath

@CachedFunction
def a(n): # a = A192386
    if (n<5): return (0,1,0,8,8)[n]
    else: return 2*a(n-1) +12*a(n-2) -8*a(n-3) -16*a(n-4)
[a(n) for n in range(1,41)] # G. C. Greubel, Jul 10 2023

Formula

From Colin Barker, May 11 2014: (Start)

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

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

From G. C. Greubel, Jul 10 2023: (Start)

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

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