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A059853
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Period of continued fraction for sqrt(n^2+3), n >= 2.
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2
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4, 2, 6, 4, 2, 6, 10, 2, 12, 16, 2, 16, 20, 2, 10, 10, 2, 12, 10, 2, 28, 10, 2, 26, 16, 2, 18, 48, 2, 34, 12, 2, 26, 32, 2, 32, 32, 2, 20, 70, 2, 56, 34, 2, 24, 22, 2, 54, 52, 2, 70, 16, 2, 18, 38, 2, 16, 36, 2, 12, 72, 2, 114, 30, 2, 64, 32, 2, 52, 54, 2, 22, 92, 2, 154, 88, 2, 56
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OFFSET
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2,1
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COMMENTS
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The old name was "Quotient cycle length of sqrt(n^2+3)." - Jianing Song, May 01 2021
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LINKS
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FORMULA
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If n is a multiple of 3 then a(n) = 2.
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EXAMPLE
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sqrt(35^2+3) = [35; [23, 2, 1, 7, 8, 1, 1, 1, 2, 2, 1, 1, 5, 3, 1, 16, 1, 3, 5, 1, 1, 2, 2, 1, 1, 1, 8, 7, 1, 2, 23, 70], so a(35) = 32.
sqrt(36^2+3) = [36; 24, 72], so a(36) = 2.
sqrt(37^2+3) = [37; 24, 1, 2, 7, 1, 8, 2, 1, 1, 1, 2, 2, 5, 1, 3, 18, 3, 1, 5, 2, 2, 1, 1, 1, 2, 8, 1, 7, 2, 1, 24, 74], so a(37) = 32.
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MAPLE
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with(numtheory): [seq(nops(cfrac(sqrt(k^2+3), 'periodic', 'quotients')[2]), k=2..256)];
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CROSSREFS
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Period of continued fraction for sqrt(n^2+k): this sequence (k=3), A059855 (k=4), A059854 (k=5).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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