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Search: id:A107356
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| A107356 |
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Period of continued fraction for (1 + square root of n-th square-free integer)/2. |
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+0
1
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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; listen)
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OFFSET
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1,1
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REFERENCES
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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
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R. Knott, An Introduction to Continued Fractions
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EXAMPLE
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a(7)=2 because 11 is the 7-th smallest square-free 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 square-free integers.
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MATHEMATICA
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(* 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
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Cf. A035015, A005117.
Adjacent sequences: A107353 A107354 A107355 this_sequence A107357 A107358 A107359
Sequence in context: A110090 A092848 A128111 this_sequence A124725 A106522 A128175
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KEYWORD
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nonn
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AUTHOR
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S. R. Finch (Steven.Finch(AT)inria.fr), May 24 2005
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EXTENSIONS
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More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), May 27 2005
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