A192376 -id:A192376 - cp's OEIS frontend

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

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)

A192378 (A192377)/2.

Original entry on oeis.org

0, 1, 2, 10, 34, 131, 484, 1812, 6756, 25221, 94118, 351262, 1310918, 4892423, 18258760, 68142632, 254311752, 949104393, 3542105802, 13219318834, 49335169514, 184121359243, 687150267436, 2564479710524, 9570768574636, 35718594588045

Offset: 1

Clark Kimberling, Jun 29 2011

nonn

See A192377.

			(See A192377.)

Cf. A192232, A192376, A192377.

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

cp's OEIS Frontend

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

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