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

Author

David W. Wilson

Keywords

• nonn

Examples

			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
		

Links

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

Programs

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 *)

Extensions

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