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

4,2

Comments

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

Links

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

Formula

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

Example

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]]

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 *)

Crossrefs

Cf. A000051, A003285, A059866, A061682.

Sequence in context: A185088 A064947 A369894 * A306546 A138775 A209385

Adjacent sequences: A059923 A059924 A059925 * A059927 A059928 A059929

Keyword

nonn,nice

Author

Labos Elemer, Mar 01 2001

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

Status

approved