A020640 Successive record periods of continued fraction for sqrt(k) (period of continued fraction for sqrt(A013645(n+1))).

1, 2, 4, 5, 6, 8, 10, 12, 16, 18, 20, 22, 26, 34, 37, 40, 42, 44, 48, 52, 54, 60, 62, 64, 70, 76, 79, 88, 94, 96, 102, 104, 108, 114, 118, 122, 130, 136, 152, 156, 158, 170, 172, 174, 194, 196, 202, 207, 210, 217, 228, 234, 238, 239, 248, 262, 268, 280, 281

Offset: 1

Author

David W. Wilson

Keywords

• nonn
• nice

Links

• Patrick McKinley, Table of n, a(n) for n=1..512

Crossrefs

Cf. A003285, A013645.

Programs

Mathematica

per[n_] := ContinuedFraction[Sqrt[n]][[2]] // Length; record = 0; t = Reap[For[n = 1, n < 2*10^4, n++, If[ !IntegerQ[Sqrt[n]], p = per[n]; If[p > record, record = p; Print[{n, p}]; Sow[{n, p}]]]]][[2, 1]]; A020640 = t[[All, 2]] (* Jean-François Alcover, Dec 27 2012 *)
DeleteDuplicates[Table[If[IntegerQ[Sqrt[n]],Nothing,Length[ContinuedFraction[Sqrt[n]][[2]]]],{n,20000}],GreaterEqual] (* Harvey P. Dale, Dec 28 2023 *)

Formula

a(n) = A003285(A013645(n+1)). - Pontus von Brömssen, Nov 24 2024