A018906 Define the Shallit sequence S(a_0,a_1) by a_{n+2} is the least integer > a_{n+1}^2/a_n for n >= 0. This is S(2,6).

2, 6, 19, 61, 196, 630, 2026, 6516, 20957, 67403, 216786, 697242, 2242518, 7212542, 23197479, 74609345, 239963764, 771788146, 2482278709, 7983677414, 25677658524, 82586271099, 265619708576, 854304579262, 2747673800490, 8837259564290, 28423008798464

Offset

0,1

Links

Alois P. Heinz, 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.

Jeffrey Shallit, Problem B-686, Fib. Quart., 29 (1991), 85.

Index entries for Pisot sequences

Formula

a(n) = [ a(n-1)^2/a(n-2)+1 ].

Maple

a:= proc(n) option remember; `if`(n<2, [2, 6][n+1],
       1 +floor(a(n-1)^2/a(n-2)))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, May 05 2014

Mathematica

a[0]=2; a[1]=6; a[n_] := a[n] = Floor[a[n-1]^2/a[n-2]+1]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 30 2015 *)
nxt[{a_, b_}]:={b, Floor[b^2/a+1]}; NestList[nxt, {2, 6}, 30][[;; , 1]] (* Harvey P. Dale, Dec 01 2024 *)

Prog

(PARI) a1n=concat([ 2, 6 ], vector(28)); a(n)=a1n[ n+1 ]; for(n=2, 29, a1n[ n+1 ]=1+floor(a(n-1)^2/a(n-2)))

Crossrefs

There has been some confusion between A018906 and A014010. I think the descriptions are correct now, thanks to Michael Somos.

Sequence in context: A001169 A187276 A022041 * A014010 A022015 A138747

Adjacent sequences: A018903 A018904 A018905 * A018907 A018908 A018909

Keyword

nonn

Author

Simon Plouffe, R. K. Guy

Extensions

An incorrect g.f. was deleted by N. J. A. Sloane, Sep 16 2009

Status

approved