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A195622
Denominators of Pythagorean approximations to 5.
4
20, 2020, 206040, 21014040, 2143226060, 218588044060, 22293837268080, 2273752813300080, 231900493119340100, 23651576545359390100, 2412228907133538450120, 246023696951075562522120, 25092004860102573838806140, 2559138472033511455995704140
OFFSET
1,1
COMMENTS
See A195500 for a discussion and references.
FORMULA
From Colin Barker, Jun 03 2015: (Start)
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)
a(n) = (5/26)*(A097726(n) - (-1)^n). - G. C. Greubel, Feb 15 2023
MATHEMATICA
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]} &)[
p[{r, z}]] (* A195622, A195623 *)
Sqrt[a^2 + b^2] (* A097727 *)
(* by Peter J. C. Moses, Sep 02 2011 *)
LinearRecurrence[{101, 101, -1}, {20, 2020, 206040}, 20] (* Harvey P. Dale, Oct 17 2021 *)
PROG
(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)
[(5/26)*(A097726(n) - (-1)^n) for n in range(1, 41)] # G. C. Greubel, Feb 15 2023
CROSSREFS
KEYWORD
nonn,easy,frac
AUTHOR
Clark Kimberling, Sep 22 2011
STATUS
approved