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

0,3

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.

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.: 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

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

Mathematica

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

Prog

(Magma) R<x>:=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
(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

Crossrefs

Cf. A000045, A163762, A192232, A192428.

Sequence in context: A034960 A240372 A157493 * A354172 A280434 A163697

Adjacent sequences: A192426 A192427 A192428 * A192430 A192431 A192432

Keyword

nonn

Author

Clark Kimberling, Jun 30 2011

Status

approved