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A195614
Denominators a(n) of Pythagorean approximations b(n)/a(n) to 2.
3
8, 136, 2448, 43920, 788120, 14142232, 253772064, 4553754912, 81713816360, 1466294939560, 26311595095728, 472142416783536, 8472251907007928, 152028391909359160, 2728038802461456960, 48952670052396866112, 878420022140682133064
OFFSET
1,1
COMMENTS
See A195500 for a discussion and references.
FORMULA
From Colin Barker, Jun 04 2015: (Start)
G.f.: 8*x / ((x+1)*(x^2-18*x+1)).
a(n) = 17*a(n-1) + 17*a(n-2) - a(n-3). (End)
a(n) = (-4*(-1)^n - (-2+sqrt(5))*(9+4*sqrt(5))^(-n) + (2+sqrt(5))*(9+4*sqrt(5))^n)/10. - Colin Barker, Mar 04 2016
a(n) = A014445(n) * A014445(n+1) / 2. - Diego Rattaggi, Jun 01 2020
a(n) is the numerator of continued fraction [4, ..., 4, 8, 4, ..., 4] with (n-1) 4's before and after the middle 8. - Greg Dresden and Hexuan Wang, Aug 30 2021
MATHEMATICA
r = 2; z = 32;
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}]] (* A195614, A195615 *)
Sqrt[a^2 + b^2] (* A007805 *)
(* Peter J. C. Moses, Sep 02 2011 *)
PROG
(PARI) Vec(8*x/((x+1)*(x^2-18*x+1)) + O(x^50)) \\ Colin Barker, Jun 04 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Sep 22 2011
STATUS
approved