login
A192347
Coefficient of x in the reduction (by x^2->x+1) of polynomial p(n,x) identified in Comments.
2
0, 1, 2, 11, 32, 125, 418, 1511, 5248, 18601, 65250, 230099, 809248, 2849989, 10030018, 35311375, 124293632, 437545489, 1540200002, 5421774299, 19085364000, 67183428301, 236495292002, 832498651511, 2930516834432, 10315851565625
OFFSET
1,3
COMMENTS
To define the polynomials p(n,x), let d=sqrt(x+2); 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.
FORMULA
Conjecture: a(n) = 2*a(n-1)+6*a(n-2)-2*a(n-3)-a(n-4). G.f.: x^2*(x^2+1) / (x^4+2*x^3-6*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)=2+x+x^2 -> 3+2x
p(3,x)=6x+3x^2+x^3 -> 4+11x.
From these, we read
A192346=(1,0,3,4,...) and A192347=(1,1,2,11...)
MATHEMATICA
(See A192346.)
CROSSREFS
Sequence in context: A094792 A173707 A332612 * A031400 A085786 A034128
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jun 28 2011
STATUS
approved