A192349 Coefficient of x in the reduction (by x^2->x+1) of polynomial p(n,x) identified in Comments.

0, 1, 2, 14, 40, 180, 616, 2456, 8960, 34384, 128160, 485728, 1823360, 6882368, 25896064, 97614720, 367575040, 1384954112, 5216465408, 19651804672, 74025216000, 278859191296, 1050447030272, 3957059508224, 14906157629440, 56151566438400

Offset

1,3

Comments

To define the polynomials p(n,x), let d=sqrt(x+3); then p(n,x)=(1/2)((x+d)^n+(x-d)^n). These are similar to polynomials at A161516. 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..26.

Formula

Conjecture: a(n) = 2*a(n-1)+8*a(n-2)-4*a(n-3)-4*a(n-4). G.f.: x^2*(2*x^2+1) / (4*x^4+4*x^3-8*x^2-2*x+1). [Colin Barker, Jan 17 2013]

Example

The first four polynomials p(n,x) and their reductions are as follows:
p(0,x)=1 -> 1
p(1,x)=x -> x
p(2,x)=3+x+x^2 -> 4+2x
p(3,x)=9x+3x^2+x^3 -> 4+14x.
From these, we read
A192348=(1,0,3,4,...) and A192349=(0,1,2,14...)

Mathematica

(See A192348.)

Crossrefs

Cf. A192232, A192348.

Sequence in context: A216528 A329735 A290124 * A230801 A231081 A364218

Adjacent sequences: A192346 A192347 A192348 * A192350 A192351 A192352

Keyword

nonn

Author

Clark Kimberling, Jun 28 2011

Status

approved