A192380 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

0, 2, 4, 20, 60, 230, 776, 2792, 9720, 34410, 120780, 425788, 1497716, 5274190, 18562320, 65348560, 230024944, 809742418, 2850375060, 10033806180, 35320352940, 124333050422, 437670231064, 1540664252600, 5423363437800, 19091038878650, 67203259647836

Offset

1,2

Comments

The polynomial p(n,x) is defined by ((x+d)^n-(x-d)^n)/(2d), where d=sqrt(x+2). 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..27.

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). G.f.: 2*x^2 / (x^4+2*x^3-6*x^2-2*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)=2+x+3x^2 -> 5+4x
p(3,x)=8x+4x^2+4x^3 -> 8+20x
p(4,x)=4+4x+21x^2+10x^3+5x^4 -> 45+60x.
From these, read
A192379=(1,0,5,8,45,...) and A192380=(0,2,4,20,60,...)

Mathematica

(See A192379.)

Crossrefs

Cf. A192232, A192379, A192381.

Sequence in context: A353342 A027741 A137697 * A009336 A274520 A337616

Adjacent sequences: A192377 A192378 A192379 * A192381 A192382 A192383

Keyword

nonn,easy,changed

Author

Clark Kimberling, Jun 29 2011

Status

approved