A031416 - OEIS

A031416

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.

89, 149, 193, 218, 250, 277, 337, 493, 521, 569, 653, 709, 914, 1009, 1018, 1037, 1385, 1465, 1553, 1597, 1618, 1754, 1781, 1898, 1921, 1973, 1994, 2069, 2129, 2146, 2293, 2378, 2389, 2441, 2474, 2561, 2725, 2741, 2777, 2897, 2957, 2986, 3170, 3229, 3265

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OFFSET

1,1

LINKS

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

MATHEMATICA

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 *)

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 *)

CROSSREFS

Cf. A031404-A031423.

Subsequence of A003814.

Sequence in context: A260557 A141938 A267819 * A340307 A353249 A247114

Adjacent sequences: A031413 A031414 A031415 * A031417 A031418 A031419

KEYWORD

nonn

AUTHOR

David W. Wilson

EXTENSIONS

Initial 10 removed by T. D. Noe, Apr 03 2014

STATUS

approved