A192473 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x)=1+x^n+x^(2n+2).

4, 9, 23, 58, 149, 385, 1000, 2605, 6799, 17766, 46457, 121537, 318044, 832417, 2178919, 5703874, 14931949, 39090753, 102338336, 267921061, 701419679, 1836329614, 4807555633, 12586315393, 32951355124, 86267692665, 225851630135

Offset: 1

Author

Clark Kimberling, Jul 01 2011

Keywords

• nonn

Comments

For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.

Examples

			The first four polynomials p(n,x) and their reductions are as follows:
p(1,x)=1+x+x^4 -> 3+4x
p(2,x)=1+x^2+x^6 -> 7+9x
p(3,x)=1+x^3+x^8 -> 15+23x
p(4,x)=1+x^4+x^10 -> 37+58x.
From these, read
A192472=(3,7,15,37,...) and A192473=(4,9,23,58,...)
		

Crossrefs

Cf. A192232, A192472.

Programs

Mathematica

(See A192472.)

Formula

Conjectures from Colin Barker, Jun 07 2019: (Start)

G.f.: x*(4 - 7*x - x^2 + x^3) / ((1 - 3*x + x^2)*(1 - x - x^2)).

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

(End)