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A195616
Denominators of Pythagorean approximations to 3.
5
12, 444, 16872, 640680, 24328980, 923860548, 35082371856, 1332206269968, 50588755886940, 1921040517433740, 72948950906595192, 2770139093933183544, 105192336618554379492, 3994538652411133237140, 151687276455004508631840, 5760121966637760194772768
OFFSET
1,1
COMMENTS
See A195500 for a discussion and references.
FORMULA
From Colin Barker, Jun 04 2015: (Start)
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)
From G. C. Greubel, Feb 13 2023: (Start)
a(n) = (3/10)*(A097314(n) + (-1)^n).
a(n) = (1/20)*(A085447(2*n+1) - 6*(-1)^n). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(5/2) - 3/2 = 10 * A020797 - 3/2. - Amiram Eldar, Dec 18 2025
MATHEMATICA
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]} &)[
p[{r, z}]] (* A195616, A195617 *)
Sqrt[a^2 + b^2] (* A097315 *)
(* Peter J. C. Moses, Sep 02 2011 *)
Table[(1/20)*(LucasL[2*n+1, 6] -6*(-1)^n), {n, 40}] (* G. C. Greubel, Feb 13 2023 *)
PROG
(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)
[(A085447(2*n+1) - 6*(-1)^n)/20 for n in range(1, 41)] # G. C. Greubel, Feb 13 2023
CROSSREFS
KEYWORD
nonn,easy,frac
AUTHOR
Clark Kimberling, Sep 22 2011
STATUS
approved