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A192465 Constant term of the reduction by x^2->x+2 of the polynomial p(n,x)=1+x^n+x^(2n). 4
3, 9, 25, 93, 353, 1389, 5505, 21933, 87553, 349869, 1398785, 5593773, 22372353, 89483949, 357924865, 1431677613, 5726666753, 22906579629, 91626143745, 366504225453, 1466016202753, 5864063412909, 23456250855425, 93824997829293 (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.

LINKS

Table of n, a(n) for n=1..24.

FORMULA

Empirical G.f.: -x*(3*x-1)*(8*x^2-3)/((x-1)*(x+1)*(2*x-1)*(4*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 -> 3+2x

p(2,x)=1+x^2+x^4 -> 9+6x

p(3,x)=1+x^3+x^6 -> 25+24x

p(4,x)=1+x^4+x^8 -> 93+90x.

From these, read

A192465=(3,9,25,93,...) and A192466=(2,6,24,90,...)

MATHEMATICA

Remove["Global`*"];

q[x_] := x + 2; 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}]

(* A192465 *)

Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]

(* A192466 *)

Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 30}]

(* A192467 *)

CROSSREFS

Cf. A192232, A192466, A192464.

Sequence in context: A217995 A246653 A192371 * A012771 A178061 A120284

Adjacent sequences:  A192462 A192463 A192464 * A192466 A192467 A192468

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jul 01 2011

STATUS

approved

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Last modified July 31 04:58 EDT 2021. Contains 346367 sequences. (Running on oeis4.)