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Associated Mersenne numbers.
(Formerly M2217 N0879)
5

%I M2217 N0879 #59 Sep 08 2022 08:44:29

%S 0,1,3,1,3,11,9,8,27,37,33,67,117,131,192,341,459,613,999,1483,2013,

%T 3032,4623,6533,9477,14311,20829,30007,44544,65657,95139,139625,

%U 206091,300763,439521,646888,948051,1385429,2033193,2983787,4366197,6397723,9387072

%N Associated Mersenne numbers.

%C From _Peter Bala_, Sep 15 2019: (Start)

%C This is a linear divisibility sequence of order 6 (Haselgrove, p. 21). It is a particular case of a family of divisibility sequences studied by Roettger et al. The o.g.f. has the form x*d/dx(f(x)/(x^3*f(1/x))) where f(x) = x^3 - x^2 - 1.

%C More generally, if f(x) = 1 + P*x + Q*x^2 + x^3 or f(x) = -1 + P*x + Q*x^2 + x^3, where P and Q are integers, then the rational function x*d/dx(f(x)/(x^3*f(1/x))) is the generating function for a linear divisibility sequence of order 6. Cf. A001945. There are corresponding results when f(x) is a monic quartic polynomial with constant term 1. (End)

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Danny Rorabaugh, <a href="/A001351/b001351.txt">Table of n, a(n) for n = 0..6000</a>

%H Peter Bala, <a href="/A001351/a001351_1.pdf">Some linear divisibility sequences of order 6</a>

%H C. B. Haselgrove, <a href="/A001350/a001350.pdf">Associated Mersenne numbers</a>, Eureka, 11 (1949), 19-22. [Annotated and scanned copy]

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%H E. L. Roettger, H. C. Williams, and R. K. Guy, <a href="https://carma.newcastle.edu.au/meetings/alfcon/pdfs/Hugh_Williams-alfcon.pdf">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, Vol. 43, J. Borwein, I. Shparlinski, W. Zudilin (Eds.) 2013.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,-1,3,-1,1,-1).

%F a(n) = a(n-1) - a(n-2) + 3*a(n-3) - a(n-4) + a(n-5) - a(n-6) for n >= 6. - _Sean A. Irvine_, Sep 23 2015

%F a(n) = (alpha^n - 1)*(beta^n - 1)*(gamma^n - 1) where alpha, beta and gamma are the zeros of x^3 - x^2 - 1. - _Peter Bala_, Sep 15 2019

%p A001351:=z*(z^2-z+1)*(z^2+3*z+1)/(z^3+z-1)/(z^3-z^2-1); # conjectured by _Simon Plouffe_ in his 1992 dissertation

%t LinearRecurrence[{1, -1, 3, -1, 1, -1}, {0, 1, 3, 1, 3, 11}, 50] (* _Vincenzo Librandi_, Sep 23 2015 *)

%o (Magma) I:=[0,1,3,1,3,11]; [n le 6 select I[n] else Self(n-1) - Self(n-2) + 3*Self(n-3) - Self(n-4) + Self(n-5) - Self(n-6): n in [1..50]]; // _Vincenzo Librandi_, Sel 23 2015

%Y Cf. A001350, A001945.

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_, _R. K. Guy_

%E More terms from _Vincenzo Librandi_, Sep 23 2015