The OEIS is supported by the many generous donors to the OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 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 (* Jean-François 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified March 21 09:23 EDT 2023. Contains 361402 sequences. (Running on oeis4.)