A192378 -id:A192378 - cp's OEIS frontend

A192376 Constant term of the reduction by x^2->x+2 of the polynomial p(n,x) defined below in Comments.

1, 0, 7, 16, 73, 256, 975, 3616, 13521, 50432, 188247, 702512, 2621849, 9784832, 36517535, 136285248, 508623521, 1898208768, 7084211623, 26438637648, 98670339049, 368242718464, 1374300534895, 5128959421024, 19141537149297, 71437189176064

Offset: 1

Clark Kimberling, Jun 29 2011

nonn

The polynomial p(n,x) is defined by ((x+d)^n-(x-d)^n)/(2d), where d=sqrt(x+1). For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.

			The first five polynomials p(n,x) and their reductions are as follows:
p(0,x)=1 -> 1
p(1,x)=2x -> 2x
p(2,x)=4+x+3x^2 -> 7+4x
p(3,x)=16x+4x^2+4x^3 -> 16+20x
p(4,x)=16+8x+41x^2+10x^3+5x^4 -> 73+68x.
From these, read A192376=(1,0,7,16,73,...) and A192377=(0,2,4,20,68,...).

Cf. A192232, A192377, A192378.

q[x_] := x + 2; d = Sqrt[x + 1];
p[n_, x_] := ((x + d)^n - (x - d)^ n )/(2 d)   (* Cf. A162517 *)
Table[Expand[p[n, x]], {n, 1, 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, 1,  30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}] (* A192376 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}] (* A192377 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 30}] (* A192378 *)

Conjecture: a(n) = 2*a(n-1)+6*a(n-2)+2*a(n-3)-a(n-4). G.f.: x*(x-1)^2 / ((x+1)^2*(x^2-4*x+1)). - Colin Barker, May 11 2014

A192377 Coefficient of x in the reduction by x^2->x+2 of the polynomial p(n,x) defined below in Comments.

Original entry on oeis.org

0, 2, 4, 20, 68, 262, 968, 3624, 13512, 50442, 188236, 702524, 2621836, 9784846, 36517520, 136285264, 508623504, 1898208786, 7084211604, 26438637668, 98670339028, 368242718486, 1374300534872, 5128959421048, 19141537149272, 71437189176090

Offset: 1

Clark Kimberling, Jun 29 2011

The polynomial p(n,x) is defined by ((x+d)^n - (x-d)^n)/(2d), where d=sqrt(x+1). A192377=2*A192378. For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.

			The first five polynomials p(n,x) and their reductions are as follows:
  p(0,x)=1 -> 1
  p(1,x)=2x -> 2x
  p(2,x)=4+x+3x^2 -> 7+4x
  p(3,x)=16x+4x^2+4x^3 -> 16+20x
  p(4,x)=16+8x+41x^2+10x^3+5x^4 -> 73+68x.
From these, read (0,2,4,20,68,...)

Index entries for linear recurrences with constant coefficients, signature (2,6,2,-1).

Cf. A192376, A192378.

Mathematica
```
(* See A192376. *)
```

From Colin Barker, Dec 09 2012: (Start)
a(n) = 2*a(n-1) + 6*a(n-2) + 2*a(n-3) - a(n-4).
G.f.: 2*x^2 / ((x+1)^2*(x^2-4*x+1)). (End)

cp's OEIS Frontend

A192376 Constant term of the reduction by x^2->x+2 of the polynomial p(n,x) defined below in Comments.

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