A031409 Numbers k such that the continued fraction for sqrt(k) has even period 2*m and the m-th term of the periodic part is 6.

38, 46, 54, 62, 84, 93, 111, 129, 141, 148, 164, 172, 188, 204, 212, 230, 236, 244, 245, 252, 270, 295, 305, 330, 345, 355, 395, 426, 448, 469, 474, 497, 518, 553, 570, 581, 584, 602, 609, 616, 632, 644, 648, 658, 712, 721, 738, 742, 749, 763, 765, 777, 801

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.

Programs

Mathematica

n = 1; t = {}; While[Length[t] < 50, n++; If[! IntegerQ[Sqrt[n]], c = ContinuedFraction[Sqrt[n]]; len = Length[c[[2]]]; If[EvenQ[len] && c[[2, len/2]] == 6, AppendTo[t, n]]]]; t (* T. D. Noe, Apr 04 2014 *)
cf6Q[n_]:=Module[{s=Sqrt[n],cf,len},cf=If[IntegerQ[s],{1},ContinuedFraction[s][[2]]];len=Length[cf];EvenQ[len]&&cf[[len/2]]==6]; Select[Range[1000],cf6Q] (* Harvey P. Dale, Feb 04 2023 *)