A192429 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments.

0, 1, 4, 21, 76, 329, 1256, 5157, 20216, 81505, 322924, 1293189, 5144644, 20550089, 81881168, 326756661, 1302722672, 5196774145, 20723304532, 82657204533, 329642305468, 1314745861769, 5243461810232, 20912613564549, 83404589311592

Offset: 0

Author

Clark Kimberling, Jun 30 2011

Keywords

• nonn

Comments

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)^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.

Examples

			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 A192428 = (1, 1, 5, 11, 57, 185, ...) and a(n) = (0, 1, 4, 21, 76, 329, ...).
		

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).

Crossrefs

Cf. A000045, A163762, A192232, A192428.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x*(1+2*x+3*x^2)/(1-2*x-10*x^2+6*x^3+9*x^4) )); // G. C. Greubel, Jul 13 2023

Mathematica

(See A192428.)
LinearRecurrence[{2,10,-6,-9}, {0,1,4,21}, 40] (* G. C. Greubel, Jul 13 2023 *)

SageMath

@CachedFunction
def a(n): # a = A192429
    if (n<4): return (0,1,4,21)[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

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

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