A192422 - OEIS

%I #28 Jul 12 2023 01:14:57

%S 0,1,1,5,7,20,35,83,161,355,720,1541,3185,6733,14027,29500,61663,

%T 129403,270865,567911,1189440,2492905,5222449,10943813,22928815,

%U 48044900,100665083,210927155,441948689,926020171,1940274000,4065458669,8518311809

%N Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments.

%C 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.

%C Assuming the o.g.f. 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

%H G. C. Greubel, <a href="/A192422/b192422.txt">Table of n, a(n) for n = 0..1000</a>

%H H. C. Williams and R. K. Guy, <a href="http://dx.doi.org/10.1142/S1793042111004587">Some fourth-order linear divisibility sequences</a>, Intl. J. Number Theory 7 (5) (2011) 1255-1277.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-1,-1).

%F From _Colin Barker_, May 12 2014: (Start)

%F a(n) = a(n-1) + 3*a(n-2) - a(n-3) - a(n-4).

%F G.f.: x*(1 + x^2)/(1 - x - 3*x^2 + x^3 + x^4). (End)

%F From _Vladimir Kruchinin_, Mar 20 2016: (Start)

%F G.f.: ((1+x^2)/(1-x^2)) * F(x/(1-x^2)), where F(x) is g.f. of Fibonacci numbers (A000045).

%F a(n) = n*Sum_{i=0..floor((n-1)/2)} (binomial(n-i-1,i)/(n-2*i))*Fibonacci(n-2*i). (End)

%F a(n) = Sum_{j=0..n} T(n, j)*Fibonacci(j), where T(n, k) = [x^k] ((x + sqrt(x^2+4))^n + (x - sqrt(x^2+4))^n)/2^n. - _G. C. Greubel_, Jul 11 2023

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

%e   p(0,x) = 2 -> 2

%e   p(1,x) = x -> x

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

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

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

%e From these, read A192421 = (2, 0, 3, 1, 8, ...) and a(n) = (0, 1, 1, 5, 7, ...).

%t (See A192421.)

%t LinearRecurrence[{1,3,-1,-1}, {0,1,1,5}, 40] (* _G. C. Greubel_, Jul 11 2023 *)

%o (Maxima)

%o 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 */

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x*(1+x^2)/(1-x-3*x^2+x^3+x^4) )); // _G. C. Greubel_, Jul 11 2023

%o (SageMath)

%o @CachedFunction

%o def a(n): # a = A192422

%o     if (n<4): return (0,1,1,5)[n]

%o     else: return a(n-1) +3*a(n-2) -a(n-3) -a(n-4)

%o [a(n) for n in range(41)] # _G. C. Greubel_, Jul 11 2023

%Y Cf. A000045, A100047, A192232, A192421.

%K nonn

%O 0,4

%A _Clark Kimberling_, Jun 30 2011