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A059855
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Period of continued fraction for sqrt(n^2+4), n >= 1.
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2
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1, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2
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OFFSET
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1,2
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COMMENTS
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The old name was "Quotient cycle length of sqrt(n^2+4)."
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LINKS
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FORMULA
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a(n) = 2 for even n, a(n) = 5 for odd n > 1.
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EXAMPLE
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For even n, sqrt(n^2+4) = [n; n/2, 2*n], hence a(n) = 2.
For odd n > 1, sqrt(n^2+4) = [n; (n-1)/2, 1, 1, (n-1)/2, 2*n], hence a(n) = 5.
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MAPLE
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with(numtheory): [seq(nops(cfrac(sqrt(k^2+4), 'periodic', 'quotients')[2]), k=1..100)];
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MATHEMATICA
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a[n_] := Length @ ContinuedFraction[Sqrt[n^2 + 4]][[2]]; Array[a, 100] (* Amiram Eldar, May 13 2020 *)
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CROSSREFS
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Period of continued fraction for sqrt(n^2+k): A059853 (k=3), this sequence (k=4), A059854 (k=5).
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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