A001351 Associated Mersenne numbers. (Formerly M2217 N0879)

0, 1, 3, 1, 3, 11, 9, 8, 27, 37, 33, 67, 117, 131, 192, 341, 459, 613, 999, 1483, 2013, 3032, 4623, 6533, 9477, 14311, 20829, 30007, 44544, 65657, 95139, 139625, 206091, 300763, 439521, 646888, 948051, 1385429, 2033193, 2983787, 4366197, 6397723, 9387072

Offset

0,3

Comments

From Peter Bala, Sep 15 2019: (Start)

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.

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)

References

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

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

Links

Danny Rorabaugh, Table of n, a(n) for n = 0..6000

Peter Bala, Some linear divisibility sequences of order 6

C. B. Haselgrove, Associated Mersenne numbers, Eureka, 11 (1949), 19-22. [Annotated and scanned copy]

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

E. L. Roettger, H. C. Williams, and 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, Vol. 43, J. Borwein, I. Shparlinski, W. Zudilin (Eds.) 2013.

Index entries for linear recurrences with constant coefficients, signature (1,-1,3,-1,1,-1).

Formula

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

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

Maple

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

Mathematica

LinearRecurrence[{1, -1, 3, -1, 1, -1}, {0, 1, 3, 1, 3, 11}, 50] (* Vincenzo Librandi, Sep 23 2015 *)

Prog

(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

Crossrefs

Cf. A001350, A001945.

Sequence in context: A126970 A204134 A233168 * A216021 A327149 A351372

Adjacent sequences: A001348 A001349 A001350 * A001352 A001353 A001354

Keyword

nonn,easy

Author

N. J. A. Sloane, R. K. Guy

Extensions

More terms from Vincenzo Librandi, Sep 23 2015

Status

approved