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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.

%I A031509 #20 Aug 19 2021 23:27:59
%S A031509 123,127,131,139,151,163,167,488,512,520,544,608,640,672,1095,1167,
%T A031509 1383,1455,1515,1944,2008,2136,2264,2456,2648,2696,3035,3115,3215,
%U A031509 3235,3415,3515,3635,3715,3735,3835,3935,4115,4135,4215,4368,4944,5496,5943,5971
%N 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.
%H A031509 Robert Israel, <a href="/A031509/b031509.txt">Table of n, a(n) for n = 1..1000</a>
%e A031509 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
%p A031509 # Maple 2016 or later.
%p A031509 filter:= proc(n) uses NumberTheory; local R;
%p A031509   if issqr(n) then return false fi;
%p A031509   R:= Term(ContinuedFraction(sqrt(n)),periodic)[2];
%p A031509   nops(R)::even and R[nops(R)/2] = 11
%p A031509 end proc:
%p A031509 select(filter, [$2..10000]); # _Robert Israel_, Jun 07 2019
%t A031509 okQ[k_] := Module[{c, lc}, If[IntegerQ[Sqrt[k]], False,
%t A031509      c = ContinuedFraction[Sqrt[k]]; lc = Length[c[[2]]];
%t A031509      EvenQ[lc] && c[[2, lc/2]] == 11]];
%t A031509 Select[Range[10000], okQ] (* _Jean-François Alcover_, Jul 09 2021 *)
%K A031509 nonn
%O A031509 1,1
%A A031509 _David W. Wilson_
%E A031509 Definitions of A031509-A031598 clarified by _N. J. A. Sloane_, Aug 17 2021