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A192425 Coefficient of x in the reduction by x^2->x+2 of the polynomial p(n,x) defined below in Comments. 4
0, 1, 1, 6, 9, 31, 60, 169, 369, 954, 2201, 5479, 12960, 31721, 75881, 184326, 443169, 1072871, 2585340, 6249329, 15074649, 36413754, 87877681, 212208719, 512231040, 1236774481, 2985612241, 7208270406, 17401713849, 42012408751 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

The polynomial p(n,x) is defined by ((x+d)/2)^n+((x-d)/2)^n, where d=sqrt(x^2+4).  For an introduction to reductions of polynomials by substitutions such as x^2->x+2, see A192232.

LINKS

Table of n, a(n) for n=0..29.

H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.

H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume

FORMULA

Empirical G.f.: x*(x^2+1)/((x^2-x-1)*(x^2+2*x-1)). [Colin Barker, Nov 13 2012]

From Peter Bala, Mar 26 2015: (Start)

The following remarks assume the o.g.f. for this sequence is

x*(x^2 + 1)/((x^2 - x - 1)*(x^2 + 2*x - 1)) as conjectured above.

This sequence is a fourth-order linear divisibility sequence. It is the case P1 = 1, P2 = -2, Q = -1 of the 3-parameter family of divisibility sequences found by Williams and Guy.

exp( Sum_{n >= 1} 3*a(n)*x^n/n ) = 1 + Sum_{n >= 1} 3*Pell(n)*x^n.

exp( Sum_{n >= 1} (-3)*a(n)*x^n/n ) = 1 + Sum_{n >= 1} 3*Fibonacci(n)*(-x)^n. Cf. A002878. (End)

EXAMPLE

(See A192423.)

MATHEMATICA

(See A192423.)

CROSSREFS

Cf. A192232, A192423.

Cf. A000045, A000129, A002878.

Sequence in context: A178597 A179908 A180325 * A219687 A147415 A217048

Adjacent sequences:  A192422 A192423 A192424 * A192426 A192427 A192428

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jun 30 2011

STATUS

approved

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Last modified September 24 03:21 EDT 2020. Contains 337315 sequences. (Running on oeis4.)