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A327542 A linear divisibility sequence of order 8. 1
1, 2, 16, 36, 171, 512, 2087, 6984, 26512, 92682, 341573, 1216512, 4429309, 15898766, 57595536, 207410832, 749793263, 2703799808, 9765692771, 35235657396, 127218945296, 459128080534, 1657436539337, 5982212358144, 21594204190521 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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).

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.

There are corresponding results for certain cubic polynomials - see A001945. See also A327541.

LINKS

Table of n, a(n) for n=1..25.

P. Bala, Some linear divisibility sequences of order 8

E. L. Roettger, H. C. Williams, R. K. Guy, Some extensions of the Lucas functions, Number Theory and Related Fields: In Memory of Alf van der Poorten, Series: Springer Proceedings in Mathematics & Statistics 43, 271-311 (2013), chapter 5.

Index entries for linear recurrences with constant coefficients, signature (2,7,-6,4,6,7,-2,-1).

FORMULA

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.

a(2*n+1) = (-1/4) * Sum_{i = 1..4} (alpha(i)^(2*n+1) + 1/alpha(i)^(2*n+1)).

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.

a(2*n+1)^2 = (1/16) * Product_{i = 1..6} (1 + beta(i)^(2*n+1)).

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).

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)).

MATHEMATICA

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];

a /@ Range[25] (* Jean-Fran├žois Alcover, Nov 11 2019 *)

CROSSREFS

Cf. A001351, A001945, A327541.

Sequence in context: A018951 A019064 A123135 * A166154 A034507 A211620

Adjacent sequences:  A327539 A327540 A327541 * A327543 A327544 A327545

KEYWORD

nonn,easy

AUTHOR

Peter Bala, Sep 23 2019

STATUS

approved

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Last modified August 11 12:12 EDT 2022. Contains 356065 sequences. (Running on oeis4.)