A031417 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 4.

274, 370, 481, 797, 953, 1069, 1249, 1313, 1378, 1381, 1514, 1657, 1658, 1733, 1889, 2125, 2297, 2377, 2554, 2557, 2833, 2834, 2929, 2941, 3226, 3329, 3338, 3433, 3541, 3761, 3874, 3989, 4093, 4106, 4441, 4442, 4561, 4682, 4685, 4933, 4937, 5197, 5450

Offset

1,1

Links

T. D. Noe, Table of n, a(n) for n = 1..999

Example

The simple continued fraction for sqrt(274) = [16; 1, 1, 4, 4, 1, 1, 32, ...] with odd period 7 and central term 4. Another example is sqrt(481) = [21; 1, 13, 1, 1, 1, 4, 4, 1, 1, 1, 13, 1, 42, ...] with odd period 13 and central term 4. - Michael Somos, Apr 03 2014

Mathematica

n = 1; t = {}; While[Length[t] < 50, n++; If[! IntegerQ[Sqrt[n]], c = ContinuedFraction[Sqrt[n]]; len = Length[c[[2]]]; If[OddQ[len] && c[[2, (len + 1)/2]] == 4, AppendTo[t, n]]]]; t (* T. D. Noe, Apr 03 2014 *)
cf4Q[n_]:=Module[{s=Sqrt[n], cf, len}, cf=If[IntegerQ[s], {1, 1}, ContinuedFraction[ s][[2]]]; len=Length[cf]; OddQ[len]&&cf[[(len+1)/2]] == cf[[(len-1)/2]]==4]; Select[Range[5500], cf4Q] (* Harvey P. Dale, Jul 28 2021 *)

Crossrefs

Cf. A031404-A031423.

Subsequence of A003814.

Sequence in context: A307537 A295455 A174771 * A361797 A260289 A322254

Adjacent sequences: A031414 A031415 A031416 * A031418 A031419 A031420

Keyword

nonn

Author

David W. Wilson

Extensions

a(1) corrected by T. D. Noe, Apr 03 2014

Status

approved