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

Author

Clark Kimberling, Jun 29 2011

Keywords

• nonn
• easy

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.

Examples

			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,...)
		

Links

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

Crossrefs

Cf. A192232, A192379, A192381.

Programs

Mathematica

(See A192379.)

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]