A192466 Coefficient of x in the reduction by x^2->x+2 of the polynomial p(n,x)=1+x^n+x^(2n).

2, 6, 24, 90, 352, 1386, 5504, 21930, 87552, 349866, 1398784, 5593770, 22372352, 89483946, 357924864, 1431677610, 5726666752, 22906579626, 91626143744, 366504225450, 1466016202752, 5864063412906, 23456250855424, 93824997829290

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.

Yuanan Diao, Michael Finney, and Dawn Ray, The number of oriented rational links with a given deficiency number, arXiv:2007.02819 [math.GT], 2020. See p. 16.

Formula

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

Conjectures from Colin Barker, Feb 14 2017: (Start)

a(n) = (-1 - (-1)^n + 2^n + 4^n) / 3.

a(n) = 6*a(n-1) - 7*a(n-2) - 6*a(n-3) + 8*a(n-4) for n>4.

(End)

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

(See A192465.)

Crossrefs

Cf. A192232, A192465, A192467.

Sequence in context: A293774 A226037 A003450 * A367274 A361752 A115220

Adjacent sequences: A192463 A192464 A192465 * A192467 A192468 A192469

Keyword

nonn

Author

Clark Kimberling, Jul 01 2011

Status

approved