OFFSET
1,3
COMMENTS
1
FORMULA
Empirical G.f.: -x*(x^3-x^2-2*x+1)/((x^2-3*x+1)*(x^2-x-1)). [Colin Barker, Sep 11 2012]
EXAMPLE
For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.
The first six polynomials p(n,x) and their reductions are as follows:
p(0,x)=1 -> 1
p(1,x)=x -> x
p(2,x)=1+x^2 -> 2+x
p(3,x)=3x+x^3 -> 1+5x
p(4,x)=1+6x^2+x^4 -> 9+9x
p(5,x)=5x+10x^3+x^5 -> 13+30x.
From these, we read
MATHEMATICA
q[x_] := x + 1; d = 1;
p[n_, x_] := ((x + d)^n + (x - d)^n )/
2 (* similar to polynomials defined at A161516 *)
Table[Expand[p[n, x]], {n, 0, 6}]
reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0, 30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]
(* A192352 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
(* A049602 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jun 29 2011
STATUS
approved