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A107356
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Period of continued fraction for (1 + square root of n-th squarefree integer)/2.
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0
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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REFERENCES
| K. S. Williams and N. Buck, Comparison of the lengths of the continued fractions of sqrt(D) and 1+sqrt(D))/2, Proc. Amer. Math. Soc. 120 (1994) 995-1002.
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LINKS
| R. Knott, An Introduction to Continued Fractions
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FORMULA
| a(n) = A146326(A005117(n+1)). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 24 2009]
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EXAMPLE
| a(7)=2 because 11 is the 7-th 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.
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MATHEMATICA
| (* first do *) Needs["NumberTheory`NumberTheoryFunctions`"] (* then *) s = Drop[ Select[ Range[162], SquareFreeQ[ # ] &], 1]; Length[ ContinuedFraction[ # ][[2]]] & /@ ((1 + Sqrt[s])/2) (from Robert G. Wilson v (rgwv(AT)rgwv.com), May 27 2005)
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CROSSREFS
| Cf. A035015, A005117.
Sequence in context: A196831 A092848 A128111 * A124725 A106522 A128175
Adjacent sequences: A107353 A107354 A107355 * A107357 A107358 A107359
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KEYWORD
| nonn
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AUTHOR
| S. R. Finch (Steven.Finch(AT)inria.fr), May 24 2005
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EXTENSIONS
| More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), May 27 2005
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