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A192464 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x)=1+x^n+x^(2n). 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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.

LINKS

Table of n, a(n) for n=1..29.

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

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]

EXAMPLE

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,...)

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 *)

CROSSREFS

Cf. A192232, A051450.

Sequence in context: A319559 A260790 A151378 * A137568 A010355 A171880

Adjacent sequences:  A192461 A192462 A192463 * A192465 A192466 A192467

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jul 01 2011

STATUS

approved

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Last modified January 27 14:39 EST 2020. Contains 331295 sequences. (Running on oeis4.)