A107356 Period of continued fraction for (1 + square root of n-th squarefree integer)/2.

2, 2, 1, 4, 4, 2, 2, 1, 4, 2, 3, 6, 2, 6, 4, 2, 1, 2, 8, 4, 4, 2, 3, 6, 6, 5, 4, 10, 8, 4, 2, 1, 4, 6, 6, 6, 3, 4, 3, 6, 10, 4, 6, 8, 9, 6, 2, 4, 4, 2, 2, 1, 6, 2, 7, 8, 2, 12, 4, 9, 3, 6, 12, 6, 18, 6, 7, 4, 6, 7, 6, 6, 14, 4, 2, 2, 12, 10, 6, 6, 4, 10, 7, 4, 18, 4, 4, 2, 3, 6, 5, 20, 14, 8, 5, 12, 6, 10

Offset: 1

Author

Steven Finch, May 24 2005

Keywords

• nonn

Examples

			a(7) = 2 because 11 is the 7th smallest squarefree integer and (1 + sqrt 11)/2 = [2,6,3,6,3,6,3,... ] thus has an eventual period of 2. We omit 1 from the list of squarefree integers.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Ron Knott, An Introduction to Continued Fractions
• K. S. Williams and N. Buck, Comparison of the lengths of the continued fractions of sqrt(D) and (1/2)*(1+sqrt(D)), Proc. Amer. Math. Soc. 120 (1994) 995-1002.

Crossrefs

Cf. A005117, A035015, A146326.

Programs

Mathematica

(* first do *) Needs["NumberTheory`NumberTheoryFunctions`"] (* then *) s = Drop[ Select[ Range[162], SquareFreeQ[ # ] &], 1]; Length[ ContinuedFraction[ # ][[2]]] & /@ ((1 + Sqrt[s])/2) (* Robert G. Wilson v, May 27 2005 *)
Length[ContinuedFraction[(Sqrt[#]+1)/2][[2]]]&/@Select[Range[ 2,200], SquareFreeQ] (* Harvey P. Dale, Aug 16 2021 *)

Formula

a(n) = A146326(A005117(n+1)). - R. J. Mathar, Sep 24 2009

Extensions

More terms from Robert G. Wilson v, May 27 2005