A031423 Numbers k such that the continued fraction for sqrt(k) has odd period and if the last term of the periodic part is deleted then there are a pair of central terms both equal to 10.

701, 1418, 1493, 2197, 2290, 3257, 4793, 6154, 6466, 8389, 8753, 9577, 9965, 10765, 11257, 11677, 12541, 14218, 14929, 15413, 15658, 16001, 16501, 17009, 17786, 18049, 18314, 18581, 19121, 21577, 22157, 22745, 24557, 24677, 25805, 26561, 27530, 28517

Offset: 1

Author

David W. Wilson

Keywords

• nonn

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe and Georg Fischer)

Crossrefs

Cf. A031404-A031422.

Subsequence of A003814.

Programs

Mathematica

n = 1; t = {}; While[Length[t] < 50, n++; If[! IntegerQ[Sqrt[n]], c = ContinuedFraction[Sqrt[n]]; len = Length[c[[2]]]; If[OddQ[len] && c[[2, (len + 1)/2]] == 10 && c[[2, (len + 1)/2 - 1]] == 10, AppendTo[t, n]]]]; t (* T. D. Noe, Apr 04 2014; corrected by Georg Fischer, Jun 23 2019 *)

Python

from sympy.ntheory.continued_fraction import continued_fraction_periodic
A031423_list = []
for n in range(1,10**4):
    cf = continued_fraction_periodic(0,1,n)
    if len(cf) > 1 and len(cf[1]) > 1 and len(cf[1]) % 2 and cf[1][len(cf[1])//2] == 10:
        A031423_list.append(n) # Chai Wah Wu, Sep 16 2021

Extensions

a(1) corrected by T. D. Noe, Apr 04 2014

a(1) = 26 removed by Georg Fischer, Jun 23 2019