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A195499
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Denominators a(n) of Pythagorean approximations b(n)/a(n) to sqrt(3).
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5
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3, 8, 33, 120, 451, 1680, 6273, 23408, 87363, 326040, 1216801, 4541160, 16947843, 63250208, 236052993, 880961760, 3287794051, 12270214440, 45793063713, 170902040408, 637815097923, 2380358351280, 8883618307201, 33154114877520
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OFFSET
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1,1
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COMMENTS
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See A195500 for a discussion and references.
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LINKS
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FORMULA
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Empirical G.f.: x*(3-x)/(1-3*x-3*x^2+x^3). - Colin Barker, Jan 04 2012
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EXAMPLE
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From the Pythagorean triples (3,4,5), (8,15,17),(33,56,65), (120,209,241), (451,780,901), read the first five best approximating fractions b(n)/a(n):
4/3, 15/8, 56/33, 209/120, 780/451.
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MATHEMATICA
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r = Sqrt[3]; z = 25;
p[{f_, n_}] := (#1[[2]]/#1[[
1]] &)[({2 #1[[1]] #1[[2]], #1[[1]]^2 - #1[[
2]]^2} &)[({Numerator[#1], Denominator[#1]} &)[
Array[FromContinuedFraction[
ContinuedFraction[(#1 + Sqrt[1 + #1^2] &)[f], #1]] &, {n}]]]];
{a, b} = ({Denominator[#1], Numerator[#1]} &)[
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CROSSREFS
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KEYWORD
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nonn,easy,frac
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AUTHOR
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STATUS
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approved
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