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A327542 A linear divisibility sequence of order 8.

%I A327542 #12 Nov 11 2019 09:34:37
%S A327542 1,2,16,36,171,512,2087,6984,26512,92682,341573,1216512,4429309,
%T A327542 15898766,57595536,207410832,749793263,2703799808,9765692771,
%U A327542 35235657396,127218945296,459128080534,1657436539337,5982212358144,21594204190521
%N A327542 A linear divisibility sequence of order 8.
%C A327542 Let f(x) = 1 + P*x + Q*x^2 + R*x^3 + x^4 be a monic quartic polynomial with integer coefficients. Let g(x) = x^4*f(1/x) = 1 + R*x + Q*x^2 + P*x^3 + x^4 be the reciprocal polynomial of f(x). Then the rational function x*d/dx( log(f(x)/g(-x)) ) is the generating function for a divisibility sequence satisfying a linear recurrence equation of order 8. Here we take f(x) = 1 + x - 2*x^2 + 3*x^3 + x^4 (and normalize the resulting divisibility sequence by removing a common factor of 4 from the terms of the sequence).
%C A327542 Roettger et al. constructed a 5-parameter family U_n(P1,P2,P3,P4,Q) of linear divisibility sequences of order 8. This sequence is a particular case of their result with parameters P1 = 2, P2 = -3, P3 = 0, P4 = -16 and Q = -1.
%C A327542 There are corresponding results for certain cubic polynomials - see A001945. See also A327541.
%H A327542 P. Bala, <a href="/A327541/a327541.pdf">Some linear divisibility sequences of order 8</a>
%H A327542 E. L. Roettger, H. C. Williams, R. K. Guy, <a href="https://doi.org/10.1007/978-1-4614-6642-0_15">Some extensions of the Lucas functions</a>, Number Theory and Related Fields: In Memory of Alf van der Poorten, Series: Springer Proceedings in Mathematics & Statistics 43, 271-311 (2013), chapter 5.
%H A327542 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,7,-6,4,6,7,-2,-1).
%F A327542 a(2*n) = (1/4) * Sum_{i = 1..4} (alpha(i)^(2*n) - 1/alpha(i)^(2*n)), where alpha(i), 1 <= i <= 4, are the zeros of the quartic polynomial 1 + x - 2*x^2 + 3*x^3 + x^4.
%F A327542 a(2*n+1) = (-1/4) * Sum_{i = 1..4} (alpha(i)^(2*n+1) + 1/alpha(i)^(2*n+1)).
%F A327542 a(2*n)^2 = (-1/16) * Product_{i = 1..6} (1 - beta(i)^(2*n)), where beta(i), 1 <= i <= 6, are the zeros of the sextic polynomial x^6 + 2*x^5 + 2*x^4 - 14*x^3 + 2*x^2 + 2*x + 1.
%F A327542 a(2*n+1)^2 = (1/16) * Product_{i = 1..6} (1 + beta(i)^(2*n+1)).
%F A327542 a(n) = 2*a(n-1) + 7*a(n-2) - 6*a(n-3) + 4*a(n-4) + 6*a(n-5) + 7*a(n-6) - 2*a(n-7) - a(n-8).
%F A327542 O.g.f.: x*(1 + 5*x^2 - 4*x^3 - 5*x^4 - x^6)/((1 + x - 2*x^2 + 3*x^3 + x^4)*(1 - 3*x - 2*x^2 - x^3 + x^4)).
%t A327542 a[n_] := With[{m = 1 - 2 Mod[n, 2]}, (m/4)(x^n - m/x^n) /. {Roots[1 + x - 2x^2 + 3x^3 + x^4 == 0, x] // ToRules} // Total // Round];
%t A327542 a /@ Range[25] (* _Jean-François Alcover_, Nov 11 2019 *)
%Y A327542 Cf. A001351, A001945, A327541.
%K A327542 nonn,easy
%O A327542 1,2
%A A327542 _Peter Bala_, Sep 23 2019