A031405 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 2.

6, 12, 14, 20, 21, 28, 30, 33, 42, 44, 45, 52, 55, 56, 60, 70, 72, 77, 90, 95, 110, 112, 117, 126, 132, 133, 138, 153, 154, 156, 161, 165, 180, 182, 184, 189, 190, 207, 209, 210, 221, 234, 240, 248, 253, 261, 272, 275, 276, 285, 286, 297, 299, 306, 310, 315

Offset: 1

Author

David W. Wilson

Keywords

• nonn

Examples

			The c.f. for sqrt(6) is [2; 2, 4, ...] with period 2 and 1st term of the periodic part 2.
The c.f. for sqrt(14) is [3; 1, 2, 1, 6, ...] with period 4 and 2nd term of the periodic part 2.
The c.f. for sqrt(21) is [4; 1, 1, 2, 1, 1, 8, ...] with period 6 and 3rd term of the periodic part 2.
		

Links

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

Crossrefs

Cf. A031404-A031423.

Programs

Maple

filter:= proc(n) local P,l;
  if issqr(n) then return false fi;
  P:= numtheory:-cfrac(sqrt(n),'periodic','quotients')[2];
  l:= nops(P);
  if l::odd then false
  else P[l/2] = 2
  fi
end proc:
select(filter, [$1..1000]); # Robert Israel, Apr 14 2016

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]] == 2, AppendTo[t, n]]]]; t