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

Author

Georg Fischer, Jun 24 2019

Keywords

• nonn

Examples

			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.
		

Links

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

Crossrefs

Cf. A013642, A308778.

Programs

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 *)