A031416 - OEIS

%I #20 Aug 18 2021 00:33:16

%S 89,149,193,218,250,277,337,493,521,569,653,709,914,1009,1018,1037,

%T 1385,1465,1553,1597,1618,1754,1781,1898,1921,1973,1994,2069,2129,

%U 2146,2293,2378,2389,2441,2474,2561,2725,2741,2777,2897,2957,2986,3170,3229,3265

%N Numbers k such that the continued fraction for sqrt(k) has odd period and if the last term of the periodic part is deleted then there are a pair of central terms both equal to 3.

%H T. D. Noe, <a href="/A031416/b031416.txt">Table of n, a(n) for n = 1..1000</a>

%t n = 1; t = {}; While[Length[t] < 60, n++; If[! IntegerQ[Sqrt[n]], c = ContinuedFraction[Sqrt[n]]; len = Length[c[[2]]]; If[OddQ[len] && c[[2, (len + 1)/2]] == 3, AppendTo[t, n]]]]; t (* _T. D. Noe_, Apr 03 2014 *)

%t cfo3Q[n_]:=Module[{s=Sqrt[n],cf,len},cf=If[IntegerQ[s],{0,0}, ContinuedFraction[ s ][[2]]];len=Length[cf];OddQ[len]&&cf[[ (len+1)/2]] == cf[[(len-1)/2]]==3]; Select[Range[3300],cfo3Q] (* _Harvey P. Dale_, Sep 25 2019 *)

%Y Cf. A031404-A031423.

%Y Subsequence of A003814.

%K nonn

%O 1,1

%A _David W. Wilson_

%E Initial 10 removed by _T. D. Noe_, Apr 03 2014