A059926 Length of period of the continued fraction expansion of sqrt(2^n+1).

1, 4, 1, 10, 1, 16, 1, 44, 1, 74, 1, 46, 1, 204, 1, 714, 1, 702, 1, 908, 1, 404, 1, 7754, 1, 1136, 1, 9886, 1, 8154, 1, 23578, 1, 65096, 1, 404762, 1, 23992, 1, 3514774, 1, 110124, 1, 4802160, 1, 6490450, 1, 180832, 1, 115972, 1, 770304, 1, 62665998, 1, 133093360, 1, 1019300318, 1, 60079334

Offset: 4

Author

Labos Elemer, Mar 01 2001

Keywords

• nonn
• nice

Comments

For n=1,2 a(1)=2, a(2)=1; for n=3 it is not a quadratic surd.

Examples

			For n=7 and n=8 the periods after the transient are as follows: cfrac(sqrt(2^7+1),'periodic','quotients'); gives [[11], [2, 1, 3, 1, 6, 1, 3, 1, 2, 22]] cfrac(sqrt(2^8+1),'periodic','quotients'); gives [[16], [32]]
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 4..84

Crossrefs

Cf. A000051, A003285, A059866, A061682.

Programs

Maple

with(numtheory): [seq(nops(cfrac(sqrt(2^k+1),'periodic','quotients')[2]),k=4..28)];

Mathematica

Table[Length[ContinuedFraction[Sqrt[2^n+1]][[2]]],{n,4,60}] (* Harvey P. Dale, Feb 05 2012 *)

Formula

a(n) = A003285(A000051(n)). - Michel Marcus, Sep 27 2019

Extensions

Two more terms from David W. Wilson, Jun 18 2001

Corrected and extended by Naohiro Nomoto, Nov 09 2001

a(58)-a(63) from Daniel Suteu, Jan 25 2019