

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
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OFFSET

0,4


COMMENTS

The polynomial p(n,x) is defined by ((x+d)/2)^n+((xd)/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 fourthorder linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 12551277.
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^2x1)*(x^2+2*x1)). [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 fourthorder linear divisibility sequence. It is the case P1 = 1, P2 = 2, Q = 1 of the 3parameter 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



