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

%I A107356 #33 Aug 16 2021 15:22:48
%S A107356 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,
%T A107356 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,
%U A107356 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
%N A107356 Period of continued fraction for (1 + square root of n-th squarefree integer)/2.
%H A107356 Amiram Eldar, <a href="/A107356/b107356.txt">Table of n, a(n) for n = 1..10000</a>
%H A107356 Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/cfINTRO.html">An Introduction to Continued Fractions</a>
%H A107356 K. S. Williams and N. Buck, <a href="https://doi.org/10.1090/S0002-9939-1994-1169053-7">Comparison of the lengths of the continued fractions of sqrt(D) and (1/2)*(1+sqrt(D))</a>, Proc. Amer. Math. Soc. 120 (1994) 995-1002.
%F A107356 a(n) = A146326(A005117(n+1)). - _R. J. Mathar_, Sep 24 2009
%e A107356 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.
%t A107356 (* 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 *)
%t A107356 Length[ContinuedFraction[(Sqrt[#]+1)/2][[2]]]&/@Select[Range[ 2,200], SquareFreeQ] (* _Harvey P. Dale_, Aug 16 2021 *)
%Y A107356 Cf. A005117, A035015, A146326.
%K A107356 nonn
%O A107356 1,1
%A A107356 _Steven Finch_, May 24 2005
%E A107356 More terms from _Robert G. Wilson v_, May 27 2005