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

1, 1, 5, 11, 57, 185, 829, 3067, 12801, 49633, 201413, 794747, 3190617, 12673529, 50672029, 201782923, 805529409, 3210794113, 12810136517, 51078991403, 203744818617, 812521585145, 3240726179389, 12924488375899, 51547405667265

Offset

0,3

Comments

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.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,10,-6,-9).

Formula

From Colin Barker, May 12 2014: (Start)

a(n) = 2*a(n-1) + 10*a(n-2) - 6*a(n-3) - 9*a(n-4).

G.f.: (1-x-7*x^2-3*x^3)/(1-2*x-10*x^2+6*x^3+9*x^4). (End)

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

Example

The first five polynomials p(n,x) and their reductions are as follows:
  p(0,x) = 1 -> 1
  p(1,x) = 1 + x -> 1 + x
  p(2,x) = 4 + 3*x + x^2 -> 5 + 4*x
  p(3,x) = 4 + 13*x + 6*x^2 + x^3 -> 11 + 21*x
  p(4,x) = 16 + 24*x + 29*x^2 + 10*x^3 + x^4 -> 57 + 76*x.
From these, read a(n) = (1, 1, 5, 11, 57, 185, ...) and A192429 = (0, 1, 4, 21, 76, 329, ...).

Mathematica

q[x_]:= x+1; d= Sqrt[x+4];
u[x_]:= x+d; v[x_]:= x-d;
p[n_, x_]:= (u[x]^n +v[x]^n)/2 + (u[x]^n -v[x]^n)/(2*d) (* A163762 *)
Table[Expand[p[n, x]], {n, 0, 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, 0, 30}]
Table[Coefficient[Part[t, n], x, 0], {n, 30}] (* A192428 *)
Table[Coefficient[Part[t, n], x, 1], {n, 30}] (* A192429 *)
LinearRecurrence[{2, 10, -6, -9}, {1, 1, 5, 11}, 40] (* G. C. Greubel, Jul 13 2023 *)

Prog

(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
(SageMath)
@CachedFunction
def a(n): # a = A192428
    if (n<4): return (1, 1, 5, 11)[n]
    else: return 2*a(n-1) + 10*a(n-2) - 6*a(n-3) - 9*a(n-4)
[a(n) for n in range(41)] # G. C. Greubel, Jul 13 2023

Crossrefs

Cf. A163762, A192232, A192429.

Sequence in context: A149531 A083860 A141496 * A060358 A091798 A070198

Adjacent sequences: A192425 A192426 A192427 * A192429 A192430 A192431

Keyword

nonn

Author

Clark Kimberling, Jun 30 2011

Status

approved