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A195616
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Denominators of Pythagorean approximations to 3.
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5
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12, 444, 16872, 640680, 24328980, 923860548, 35082371856, 1332206269968, 50588755886940, 1921040517433740, 72948950906595192, 2770139093933183544, 105192336618554379492, 3994538652411133237140, 151687276455004508631840
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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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a(n) = 37*a(n-1) + 37*a(n-2) - a(n-3).
G.f.: 12*x / ((1+x)*(1-38*x+x^2)). (End)
a(n) = (3/10)*(A097314(n) + (-1)^n).
a(n) = (1/20)*(A085447(2*n+1) - 6*(-1)^n). (End)
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MATHEMATICA
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r = 3; z = 20;
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]} &)[
Table[(1/20)*(LucasL[2*n+1, 6] -6*(-1)^n), {n, 40}] (* G. C. Greubel, Feb 13 2023 *)
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PROG
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(PARI) Vec(12*x/((1+x)*(1-38*x+x^2)) + O(x^20)) \\ Colin Barker, Jun 04 2015
(Magma) I:=[12, 444, 16872]; [n le 3 select I[n] else 37*Self(n-1) +37*Self(n-2) -Self(n-3): n in [1..40]]; // G. C. Greubel, Feb 13 2023
(SageMath)
A085447=BinaryRecurrenceSequence(6, 1, 2, 6)
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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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