A010911 Pisot sequence E(3,11), a(n) = floor(a(n-1)^2/a(n-2) + 1/2).

3, 11, 40, 145, 526, 1908, 6921, 25105, 91065, 330326, 1198213, 4346356, 15765820, 57188385, 207443151, 752472043, 2729490816, 9900859685, 35914032730, 130273308376, 472548850273, 1714107200301, 6217692609825, 22553841080350, 81811015661001, 296758421753528

Offset

0,1

Links

Colin Barker, Table of n, a(n) for n = 0..1000

D. W. Boyd, Some integer sequences related to the Pisot sequences, Acta Arithmetica, 34 (1979), 295-305.

D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993.

S. B. Ekhad, N. J. A. Sloane, D. Zeilberger, Automated proofs (or disproofs) of linear recurrences satisfied by Pisot Sequences, arXiv:1609.05570 [math.NT] (2016).

Index entries for linear recurrences with constant coefficients, signature (3,2,1).

Formula

Is it true that a(n+3)=3*a(n+2)+2*a(n+1)+a(n)? - Claude Lenormand (claude.lenormand(AT)free.fr), Dec 05 2001

Empirical g.f.: (3+2*x+x^2) / (1-3*x-2*x^2-x^3). - Colin Barker, Jun 05 2016

Theorem: a(n) = 3 a(n - 1) + 2 a(n - 2) + a(n - 3) for n>=3. Proved using the PtoRv program of Ekhad-Sloane-Zeilberger, and implies the above conjectures. - N. J. A. Sloane, Sep 09 2016

a(n) = A108153(n+2). - Jinyuan Wang, Mar 10 2020

Mathematica

LinearRecurrence[{3, 2, 1}, {3, 11, 40}, 30] (* Jean-François Alcover, Oct 05 2018 *)

Prog

(PARI) x='x+O('x^33); Vec((3+2*x+x^2)/(1-3*x-2*x^2-x^3)) \\ Altug Alkan, Oct 05 2018

Crossrefs

Cf. A108153.

Sequence in context: A333548 A136775 A108153 * A052941 A296221 A371872

Adjacent sequences: A010908 A010909 A010910 * A010912 A010913 A010914

Keyword

nonn,easy

Author

Simon Plouffe

Status

approved