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A192350
Constant term of the reduction (by x^2->x+1) of polynomial p(n,x) identified in Comments.
2
1, 0, 6, 4, 64, 128, 896, 2752, 14208, 52224, 238592, 946176, 4110336, 16830464, 71598080, 297140224, 1253048320, 5229707264, 21973303296, 91924463616, 385642135552, 1614916091904, 6770569248768, 28364203098112, 118885634277376
OFFSET
1,3
COMMENTS
To define the polynomials p(n,x), let d=sqrt(x+5); 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)+12*a(n-2)-8*a(n-3)-16*a(n-4). G.f.: -x*(6*x^2+2*x-1) / (16*x^4+8*x^3-12*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)=5+x+x^2 -> 6+2x
p(3,x)=15x+3x^2+x^3 -> 4+20x.
From these, we read
A192350=(1,0,6,4,...) and A192351=(0,1,2,20...)
MATHEMATICA
q[x_] := x + 1; d = Sqrt[x + 5];
p[n_, x_] := ((x + d)^n + (x - d)^n )/
2 (* similar to polynomials defined at A161516 *)
Table[Expand[p[n, x]], {n, 0, 4}]
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}]
(* A192350 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
(* A192351 *)
CROSSREFS
Sequence in context: A375789 A260716 A112521 * A308900 A354949 A239861
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jun 28 2011
STATUS
approved