A308780 First element of the periodic part of the continued fraction expansion of sqrt(k), where the period is 2.

1, 2, 1, 3, 2, 1, 4, 2, 1, 5, 2, 1, 6, 4, 3, 2, 1, 7, 2, 1, 8, 4, 2, 1, 9, 6, 3, 2, 1, 10, 5, 4, 2, 1, 11, 2, 1, 12, 8, 6, 4, 3, 2, 1, 13, 2, 1, 14, 7, 4, 2, 1, 15, 10, 6, 5, 3, 2, 1, 16, 8, 4, 2, 1, 17, 2, 1, 18, 12, 9, 6, 4, 3, 2, 1, 19, 2, 1

Offset

1,2

Links

Table of n, a(n) for n=1..78.

Georg Fischer, Table of the continued fractions of sqrt(0..20000).

Example

The continued fractions for sqrt(3..8) are:
   3 1;1,2
   4 2 (square)
   5 2;4
   6 2;2,4
   7 2;1,1,1,4
   8 2;1,4
Those for 3, 6 and 8 have a period of 2, therefore the sequence starts with 1, 2, 1.

Maple

s := proc(n) if not issqr(n) then numtheory[cfrac](sqrt(n), 'periodic', 'quotients')[2]; if nops(%) = 2 then return %[1] fi fi; NULL end:
seq(s(n), n=1..399); # Peter Luschny, Jul 01 2019

Mathematica

Reap[For[k = 3, k <= 399, k++, If[!IntegerQ[Sqrt[k]], cf = ContinuedFraction[Sqrt[k]]; If[Length[cf[[2]]] == 2, Sow[cf[[2, 1]]]]]]][[2, 1]] (* Jean-François Alcover, May 03 2024 *)
(* Second program (much simpler): *)
Table[2 a/b, {a, 1, 20}, {b, Rest@Divisors[2 a]}] // Flatten (* Jean-François Alcover, May 04 2024, after a remark by Kevin Ryde *)

Crossrefs

Cf. A013642, A308778.

Sequence in context: A083368 A112379 A361101 * A246700 A375478 A073932

Adjacent sequences: A308777 A308778 A308779 * A308781 A308782 A308783

Keyword

nonn

Author

Georg Fischer, Jun 24 2019

Status

approved