A192377 Coefficient of x in the reduction by x^2->x+2 of the polynomial p(n,x) defined below in Comments.

0, 2, 4, 20, 68, 262, 968, 3624, 13512, 50442, 188236, 702524, 2621836, 9784846, 36517520, 136285264, 508623504, 1898208786, 7084211604, 26438637668, 98670339028, 368242718486, 1374300534872, 5128959421048, 19141537149272, 71437189176090

Offset

1,2

Comments

The polynomial p(n,x) is defined by ((x+d)^n - (x-d)^n)/(2d), where d=sqrt(x+1). A192377=2*A192378. For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.

Links

Table of n, a(n) for n=1..26.

Index entries for linear recurrences with constant coefficients, signature (2,6,2,-1).

Formula

a(n) = 2*a(n-1) + 6*a(n-2) + 2*a(n-3) - a(n-4). - Colin Barker, Dec 09 2012

G.f.: 2*x^2 / ((x+1)^2*(x^2-4*x+1)). - Colin Barker, Dec 09 2012

Example

The first five polynomials p(n,x) and their reductions are as follows:
p(0,x)=1 -> 1
p(1,x)=2x -> 2x
p(2,x)=4+x+3x^2 -> 7+4x
p(3,x)=16x+4x^2+4x^3 -> 16+20x
p(4,x)=16+8x+41x^2+10x^3+5x^4 -> 73+68x.
From these, read
A192376=(1,0,7,16,73,...) and A192377=(0,2,4,20,68,...)

Mathematica

See A192376.

Crossrefs

Cf. A192376, A192378.

Sequence in context: A274520 A337616 A238229 * A286344 A279153 A204550

Adjacent sequences: A192374 A192375 A192376 * A192378 A192379 A192380

Keyword

nonn,easy

Author

Clark Kimberling, Jun 29 2011

Status

approved