A031688 Numbers k such that the continued fraction for sqrt(k) has odd period and if the last term of the periodic part is deleted the two central terms are both 100.

2501, 123953, 203482, 204385, 206197, 304825, 425273, 568741, 726193, 727898, 734738, 906665, 1119805, 1124041, 1334554, 1341493, 1346129, 1565626, 1568129, 1570634, 1573141, 1578161, 1580674, 1583189, 1590746, 1593269, 1828633, 2126605, 2138285, 2144137

Offset: 1

Author

David W. Wilson

Keywords

• nonn

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..1000

Programs

Mathematica

cf100Q[n_]:=Module[{s=Sqrt[n],cf,len},cf=If[IntegerQ[s],{1,1}, ContinuedFraction[ s][[2]]];len=Length[cf];OddQ[len] && cf[[(len+1)/2]] == 100]; Select[Range[2200000],cf100Q] (* Harvey P. Dale, May 29 2016 *)

Python

from sympy import continued_fraction_periodic
A031688_list = [n for n, s in ((i, continued_fraction_periodic(0,1,i)[-1]) for i in range(1,10**5)) if isinstance(s, list) and len(s) % 2 and s[(len(s)-1)//2] == 100] # Chai Wah Wu, Jun 11 2017

Extensions

Definition corrected and terms corrected and extended by Harvey P. Dale, May 29 2016