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%I #21 Jul 12 2023 05:52:58
%S 0,1,1,6,9,31,60,169,369,954,2201,5479,12960,31721,75881,184326,
%T 443169,1072871,2585340,6249329,15074649,36413754,87877681,212208719,
%U 512231040,1236774481,2985612241,7208270406,17401713849,42012408751
%N Coefficient of x in the reduction by x^2 -> x+2 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+2, see A192232.
%H G. C. Greubel, <a href="/A192425/b192425.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 H. C. Williams and R. K. Guy, <a href="http://www.emis.de/journals/INTEGERS/papers/a17self/a17self.Abstract.html">Some Monoapparitic Fourth Order Linear Divisibility Sequences</a> Integers, Volume 12A (2012) The John Selfridge Memorial Volume
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,-1,-1).
%F G.f.: x*(1+x^2)/((1+x-x^2)*(1-2*x-x^2)). _Colin Barker_, Nov 13 2012
%F From _Peter Bala_, Mar 26 2015: (Start)
%F The following remarks assume the o.g.f. for this sequence is x*(1 + x^2)/((1 + x - x^2)*(1 - 2*x - x^2)).
%F 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.
%F exp( Sum_{n >= 1} 3*a(n)*x^n/n ) = 1 + Sum_{n >= 1} 3*Pell(n)*x^n.
%F exp( Sum_{n >= 1} (-3)*a(n)*x^n/n ) = 1 + Sum_{n >= 1} 3*Fibonacci(n)*(-x)^n. Cf. A002878. (End)
%F From _G. C. Greubel_, Jul 12 2023: (Start)
%F a(n) = Sum_{j=0..n} T(n, j)*A001045(j), where T(n, k) = [x^k] ((x + sqrt(x^2+4))^n + (x - sqrt(x^2+4))^n)/2^n.
%F a(n) = a(n-1) + 4*a(n-2) - a(n-3) - a(n-4).
%F a(n) = (1/3)*( 2*A000129(n+1) - 2*A000129(n) - (-1)^n*A000032(n)). (End)
%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 -> 4 + x
%e p(3,x) = 3*x + x^3 -> 2 + 6*x
%e p(4,x) = 2 + 4*x^2 + x^4 -> 16 + 9*x.
%e From these, read A192423(n) = = 2*A192424(n) = (2, 0, 4, 2, 16, ...) and a(n) = (0, 1, 1, 6, 9, ...).
%t (See A192423.)
%t LinearRecurrence[{1,4,-1,-1}, {0,1,1,6}, 40] (* _G. C. Greubel_, Jul 12 2023 *)
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x*(1+x^2)/((1+x-x^2)*(1-2*x-x^2)) )); // _G. C. Greubel_, Jul 12 2023
%o (SageMath)
%o @CachedFunction
%o def a(n): # a = A192425
%o if (n<4): return (0,1,1,6)[n]
%o else: return a(n-1) +4*a(n-2) -a(n-3) -a(n-4)
%o [a(n) for n in range(41)] # _G. C. Greubel_, Jul 12 2023
%Y Cf. A000045, A000129, A001045, A002878, A192232, A192423, A192424.
%K nonn,easy
%O 0,4
%A _Clark Kimberling_, Jun 30 2011
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