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A195622
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Denominators of Pythagorean approximations to 5.
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4
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20, 2020, 206040, 21014040, 2143226060, 218588044060, 22293837268080, 2273752813300080, 231900493119340100, 23651576545359390100, 2412228907133538450120, 246023696951075562522120, 25092004860102573838806140, 2559138472033511455995704140
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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) = 101*a(n-1) + 101*a(n-2) - a(n-3).
G.f.: 20*x/((1+x)*(1-102*x+x^2)). (End)
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MATHEMATICA
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r = 5; 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]} &)[
LinearRecurrence[{101, 101, -1}, {20, 2020, 206040}, 20] (* Harvey P. Dale, Oct 17 2021 *)
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PROG
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(PARI) Vec(20*x/((x+1)*(x^2-102*x+1)) + O(x^20)) \\ Colin Barker, Jun 03 2015
(Magma) I:=[20, 2020, 206040]; [n le 3 select I[n] else 101*Self(n-1) +101*Self(n-2) -Self(n-3): n in [1..40]]; // G. C. Greubel, Feb 15 2023
(SageMath)
A097726=BinaryRecurrenceSequence(102, -1, 1, 103)
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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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