A031509 Numbers k such that the continued fraction for sqrt(k) has even period and if the last term of the periodic part is deleted the central term is 11.

123, 127, 131, 139, 151, 163, 167, 488, 512, 520, 544, 608, 640, 672, 1095, 1167, 1383, 1455, 1515, 1944, 2008, 2136, 2264, 2456, 2648, 2696, 3035, 3115, 3215, 3235, 3415, 3515, 3635, 3715, 3735, 3835, 3935, 4115, 4135, 4215, 4368, 4944, 5496, 5943, 5971

Offset

1,1

Links

Robert Israel, Table of n, a(n) for n = 1..1000

Example

The c.f. expansion of sqrt(127) is 11, [3, 1, 2, 2, 7, 11, 7, 2, 2, 1, 3, 22], [3, 1, 2, 2, 7, 11, 7, 2, 2, 1, 3, 22], ... If the 22 is deleted from the periodic part the central term is 11. - N. J. A. Sloane, Aug 17 2021

Maple

# Maple 2016 or later.
filter:= proc(n) uses NumberTheory; local R;
  if issqr(n) then return false fi;
  R:= Term(ContinuedFraction(sqrt(n)), periodic)[2];
  nops(R)::even and R[nops(R)/2] = 11
end proc:
select(filter, [$2..10000]); # Robert Israel, Jun 07 2019

Mathematica

okQ[k_] := Module[{c, lc}, If[IntegerQ[Sqrt[k]], False,
     c = ContinuedFraction[Sqrt[k]]; lc = Length[c[[2]]];
     EvenQ[lc] && c[[2, lc/2]] == 11]];
Select[Range[10000], okQ] (* Jean-François Alcover, Jul 09 2021 *)

Crossrefs

Sequence in context: A192231 A341992 A077378 * A351479 A247616 A119426

Adjacent sequences: A031506 A031507 A031508 * A031510 A031511 A031512

Keyword

nonn

Author

David W. Wilson

Extensions

Definitions of A031509-A031598 clarified by N. J. A. Sloane, Aug 17 2021

Status

approved