A031423 - OEIS

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

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OFFSET

1,1

LINKS

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

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 *)

PROG

(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

CROSSREFS

Cf. A031404-A031422.

Subsequence of A003814.

Sequence in context: A093235 A223351 A180469 * A140433 A316969 A303689