A192472 - OEIS

A192472

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

3, 7, 15, 37, 93, 239, 619, 1611, 4203, 10981, 28713, 75115, 196563, 514463, 1346647, 3525189, 9228453, 24159415, 63248571, 165584323, 433501203, 1134914117, 2971232785, 7778770707, 20365057443, 53316366199, 139583983839, 365435492581

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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..28.

FORMULA

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

MATHEMATICA

Remove["Global`*"];

q[x_] := x + 1; p[n_, x_] := 1 + x^n + x^(2 n+2);

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}]

(* A192472 *)

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

(* A192473 *)

CROSSREFS

Cf. A192232, A192473.

Sequence in context: A190571 A317881 A018020 * A147106 A078161 A216616

Adjacent sequences: A192469 A192470 A192471 * A192473 A192474 A192475

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jul 01 2011

STATUS

approved