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A308780
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First element of the periodic part of the continued fraction expansion of sqrt(k), where the period is 2.
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0
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1, 2, 1, 3, 2, 1, 4, 2, 1, 5, 2, 1, 6, 4, 3, 2, 1, 7, 2, 1, 8, 4, 2, 1, 9, 6, 3, 2, 1, 10, 5, 4, 2, 1, 11, 2, 1, 12, 8, 6, 4, 3, 2, 1, 13, 2, 1, 14, 7, 4, 2, 1, 15, 10, 6, 5, 3, 2, 1, 16, 8, 4, 2, 1, 17, 2, 1, 18, 12, 9, 6, 4, 3, 2, 1, 19, 2, 1
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OFFSET
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1,2
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LINKS
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EXAMPLE
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The continued fractions for sqrt(3..8) are:
3 1;1,2
4 2 (square)
5 2;4
6 2;2,4
7 2;1,1,1,4
8 2;1,4
Those for 3, 6 and 8 have a period of 2, therefore the sequence starts with 1, 2, 1.
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MAPLE
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s := proc(n) if not issqr(n) then numtheory[cfrac](sqrt(n), 'periodic', 'quotients')[2]; if nops(%) = 2 then return %[1] fi fi; NULL end:
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MATHEMATICA
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Reap[For[k = 3, k <= 399, k++, If[!IntegerQ[Sqrt[k]], cf = ContinuedFraction[Sqrt[k]]; If[Length[cf[[2]]] == 2, Sow[cf[[2, 1]]]]]]][[2, 1]] (* Jean-François Alcover, May 03 2024 *)
(* Second program (much simpler): *)
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CROSSREFS
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KEYWORD
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nonn,changed
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AUTHOR
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STATUS
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approved
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