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A192422 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments. 2
0, 1, 1, 5, 7, 20, 35, 83, 161, 355, 720, 1541, 3185, 6733, 14027, 29500, 61663, 129403, 270865, 567911, 1189440, 2492905, 5222449, 10943813, 22928815, 48044900, 100665083, 210927155, 441948689, 926020171, 1940274000, 4065458669, 8518311809 (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+1, see A192232.

Assuming the ogf given below, this sequence is a divisibility sequence, i.e., a(n) divides a(m) whenever n divides m. It is the case P1 = 1, P2 = -1, Q = -1 of the 3-parameter family of 4th-order linear divisibility sequences found by Williams and Guy. Cf. A100047. - Peter Bala, Aug 28 2019

LINKS

Table of n, a(n) for n=0..32.

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

FORMULA

Conjecture: a(n) = a(n-1)+3*a(n-2)-a(n-3)-a(n-4). G.f.: x*(x^2+1) / (x^4+x^3-3*x^2-x+1). - Colin Barker, May 12 2014

G.f.: F(x/(1-x^2))*(1+x^2)/(1-x^2), where F(x) is g.f. of Fibonacci numbers (A000045). - Vladimir Kruchinin, Mar 20 2016

a(n) = n*Sum_{i=0..(n-1)/2}((binomial(n-i-1,i))/(n-2*i)*f(n-2*i)), where f(n) - Fibonacci numbers (A000045). - Vladimir Kruchinin, Mar 20 2016

EXAMPLE

The first five polynomials p(n,x) and their reductions are as follows:

p(0,x)=2 -> 2

p(1,x)=x -> x

p(2,x)=2+x^2 -> 3+x

p(3,x)=3x+x^3 -> 1+5x

p(4,x)=2+4x^2+x^4 -> 8+7x.

From these, read A192421=(2,0,3,1,8,...) and A192422=(0,1,1,5,7,...).

MATHEMATICA

(See A192421.)

PROG

(Maxima)

a(n):=n*sum((binomial(n-i-1, i))/(n-2*i)*fib(n-2*i), i, 0, (n-1)/2); /* Vladimir Kruchinin, Mar 20 2016 */

CROSSREFS

Cf. A000045, A192232, A192421, A100047.

Sequence in context: A279957 A249047 A258282 * A120035 A198302 A091154

Adjacent sequences:  A192419 A192420 A192421 * A192423 A192424 A192425

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jun 30 2011

STATUS

approved

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Last modified September 26 23:32 EDT 2020. Contains 337378 sequences. (Running on oeis4.)