A031598 Numbers k such that the continued fraction for sqrt(k) has even period and if the last term of the periodic part is deleted the central term is 100.

10002, 10006, 10018, 10022, 10046, 10054, 10078, 10082, 10086, 10102, 10118, 10134, 10146, 10162, 10174, 10178, 10182, 10194, 10198, 10214, 10226, 10238, 10274, 10278, 10294, 10306, 10326, 10334, 10338, 10342, 10358, 10402, 22503, 22521, 22548, 22557

Offset: 1

Author

David W. Wilson

Keywords

• nonn

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000

Programs

Mathematica

cf100Q[n_]:=Module[{s=Sqrt[n],cf,len},cf=If[IntegerQ[s],{1},ContinuedFraction[s][[2]]];len= Length[ cf];EvenQ[len]&&cf[[(len)/2]]==100]; Select[Range[23000],cf100Q]

Python

from _future_ import division
from sympy import continued_fraction_periodic
A031598_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 == 0 and s[len(s)//2-1] == 100] # Chai Wah Wu, Jun 10 2017

Extensions

Definition corrected by Harvey P. Dale, Jul 13 2023