A192425 Coefficient of x in the reduction by x^2 -> x+2 of the polynomial p(n,x) defined below in Comments.

0, 1, 1, 6, 9, 31, 60, 169, 369, 954, 2201, 5479, 12960, 31721, 75881, 184326, 443169, 1072871, 2585340, 6249329, 15074649, 36413754, 87877681, 212208719, 512231040, 1236774481, 2985612241, 7208270406, 17401713849, 42012408751

Offset: 0

Author

Clark Kimberling, Jun 30 2011

Keywords

• nonn
• easy

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+2, see A192232.

Examples

			The first five polynomials p(n,x) and their reductions are as follows:
  p(0,x) = 2 -> 2
  p(1,x) = x -> x
  p(2,x) = 2 + x^2 -> 4 + x
  p(3,x) = 3*x + x^3 -> 2 + 6*x
  p(4,x) = 2 + 4*x^2 + x^4 -> 16 + 9*x.
From these, read A192423(n) = = 2*A192424(n) = (2, 0, 4, 2, 16, ...) and a(n) = (0, 1, 1, 6, 9, ...).
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
• H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume
• Index entries for linear recurrences with constant coefficients, signature (1,4,-1,-1).

Crossrefs

Cf. A000045, A000129, A001045, A002878, A192232, A192423, A192424.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x*(1+x^2)/((1+x-x^2)*(1-2*x-x^2)) )); // G. C. Greubel, Jul 12 2023

Mathematica

(See A192423.)
LinearRecurrence[{1,4,-1,-1}, {0,1,1,6}, 40] (* G. C. Greubel, Jul 12 2023 *)

SageMath

@CachedFunction
def a(n): # a = A192425
    if (n<4): return (0,1,1,6)[n]
    else: return a(n-1) +4*a(n-2) -a(n-3) -a(n-4)
[a(n) for n in range(41)] # G. C. Greubel, Jul 12 2023

Formula

G.f.: x*(1+x^2)/((1+x-x^2)*(1-2*x-x^2)). Colin Barker, Nov 13 2012

From Peter Bala, Mar 26 2015: (Start)

The following remarks assume the o.g.f. for this sequence is x*(1 + x^2)/((1 + x - x^2)*(1 - 2*x - x^2)).

This sequence is a fourth-order linear divisibility sequence. It is the case P1 = 1, P2 = -2, Q = -1 of the 3-parameter family of divisibility sequences found by Williams and Guy.

exp( Sum_{n >= 1} 3*a(n)*x^n/n ) = 1 + Sum_{n >= 1} 3*Pell(n)*x^n.

exp( Sum_{n >= 1} (-3)*a(n)*x^n/n ) = 1 + Sum_{n >= 1} 3*Fibonacci(n)*(-x)^n. Cf. A002878. (End)

From G. C. Greubel, Jul 12 2023: (Start)

a(n) = Sum_{j=0..n} T(n, j)*A001045(j), where T(n, k) = [x^k] ((x + sqrt(x^2+4))^n + (x - sqrt(x^2+4))^n)/2^n.

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

a(n) = (1/3)*( 2*A000129(n+1) - 2*A000129(n) - (-1)^n*A000032(n)). (End)