A010910 Pisot sequence E(4,27): a(n) = floor(a(n-1)^2/a(n-2)+1/2) for n>1, a(0)=4, a(1)=27.

4, 27, 182, 1227, 8272, 55767, 375962, 2534607, 17087452, 115197747, 776623742, 5235731187, 35297505832, 237963690927, 1604269674722, 10815436502967, 72913967391412, 491560986863307, 3313935758136902, 22341419483137947, 150618195689512192, 1015416271552762887

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 (6,5).

Formula

Conjectures from Colin Barker, Jun 05 2016: (Start)

a(n) = (((3-sqrt(14))^n*(-15+4*sqrt(14))+(3+sqrt(14))^n*(15+4*sqrt(14))))/(2*sqrt(14)).

a(n) = 6*a(n-1)+5*a(n-2) for n>1.

G.f.: (4+3*x) / (1-6*x-5*x^2).

(End)

Theorem: a(n) = 6 a(n - 1) + 5 a(n - 2) for n >= 2. Proved using the PtoRv program of Ekhad-Sloane-Zeilberger, and implies the above conjectures. - N. J. A. Sloane, Sep 09 2016

Mathematica

RecurrenceTable[{a[0] == 4, a[1] == 27, a[n] == Floor[a[n - 1]^2/a[n - 2] + 1/2]}, a, {n, 0, 25}] (* Bruno Berselli, Sep 03 2013 *)
nxt[{a_, b_}]:={b, Floor[b^2/a+1/2]}; NestList[nxt, {4, 27}, 30][[All, 1]] (* Harvey P. Dale, May 13 2018 *)

Prog

(Magma) Exy:=[4, 27]; [n le 2 select Exy[n] else Floor(Self(n-1)^2/Self(n-2)+1/2): n in [1..25]]; // Bruno Berselli, Sep 03 2013
(PARI) Vec((4+3*x)/(1-6*x-5*x^2) + O(x^25)) \\ Jinyuan Wang, Mar 10 2020

Crossrefs

Sequence in context: A061693 A005974 A289718 * A078100 A356393 A036753

Adjacent sequences: A010907 A010908 A010909 * A010911 A010912 A010913

Keyword

nonn

Author

Simon Plouffe

Extensions

More terms from Bruno Berselli, Sep 03 2013

Status

approved