A031416 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 3.

89, 149, 193, 218, 250, 277, 337, 493, 521, 569, 653, 709, 914, 1009, 1018, 1037, 1385, 1465, 1553, 1597, 1618, 1754, 1781, 1898, 1921, 1973, 1994, 2069, 2129, 2146, 2293, 2378, 2389, 2441, 2474, 2561, 2725, 2741, 2777, 2897, 2957, 2986, 3170, 3229, 3265

Offset: 1

Author

David W. Wilson

Keywords

• nonn

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A031404-A031423.

Subsequence of A003814.

Programs

Mathematica

n = 1; t = {}; While[Length[t] < 60, n++; If[! IntegerQ[Sqrt[n]], c = ContinuedFraction[Sqrt[n]]; len = Length[c[[2]]]; If[OddQ[len] && c[[2, (len + 1)/2]] == 3, AppendTo[t, n]]]]; t (* T. D. Noe, Apr 03 2014 *)
cfo3Q[n_]:=Module[{s=Sqrt[n],cf,len},cf=If[IntegerQ[s],{0,0}, ContinuedFraction[ s ][[2]]];len=Length[cf];OddQ[len]&&cf[[ (len+1)/2]] == cf[[(len-1)/2]]==3]; Select[Range[3300],cfo3Q] (* Harvey P. Dale, Sep 25 2019 *)

Extensions

Initial 10 removed by T. D. Noe, Apr 03 2014