A192464 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) = 1 + x^n + x^(2n).

2, 4, 7, 16, 38, 95, 242, 624, 1619, 4216, 11002, 28747, 75170, 196652, 514607, 1346880, 3525566, 9229063, 24160402, 63250168, 165586907, 433505384, 1134920882, 2971243731, 7778788418, 20365086100, 53316412567, 139584058864, 365435613974, 956722540271

Offset: 1

Author

Clark Kimberling, Jul 01 2011

Keywords

• nonn
• easy

Comments

For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232. The coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) = 1 + x^n + x^(2n) is 2*A051450.

Examples

			The first four polynomials p(n,x) and their reductions are as follows:
p(1,x) = 1 + x   + x^2 ->  2 +  2x
p(2,x) = 1 + x^2 + x^4 ->  4 +  4x
p(3,x) = 1 + x^3 + x^6 ->  7 + 10x
p(4,x) = 1 + x^4 + x^8 -> 16 + 24x.
From these, read
A192464 = (2, 4, 7, 16, ...) and 2*A051450 = (2, 4, 10, 24, ...).
		

Links

• Index entries for linear recurrences with constant coefficients, signature (5,-7,1,3,-1).

Crossrefs

Cf. A000045, A192232, A051450.

Programs

Mathematica

Remove["Global`*"];
q[x_] := x + 1; p[n_, x_] := 1 + x^n + x^(2 n);
Table[Simplify[p[n, x]], {n, 1, 5}]
reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
   x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[FixedPoint[Expand[#1 /. reductionRules] &, p[n, x]], {n, 1, 30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]
(* A192464 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
(* 2*A051450 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 30}]
(* A051450 *)
Table[1-Fibonacci[n]+Fibonacci[1+n]-Fibonacci[2n]+Fibonacci[1+2n], {n, 1, 29}]
(* Friedjof Tellkamp, Nov 22 2021 *)

Formula

G.f.: -x*(3*x^4-7*x^3-x^2+6*x-2)/((x-1)*(x^2-3*x+1)*(x^2+x-1)). - Colin Barker, Nov 12 2012

a(n) = 1 - Fibonacci(n) + Fibonacci(1+n) - Fibonacci(2n) + Fibonacci(1+2n). - Friedjof Tellkamp, Nov 22 2021