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

1,1

Links

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

Example

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.

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

Crossrefs

Cf. A031404-A031423.

Sequence in context: A315600 A056774 A315601 * A315602 A315603 A318145

Adjacent sequences: A031402 A031403 A031404 * A031406 A031407 A031408

Keyword

nonn

Author

David W. Wilson

Status

approved