This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
For r=sqrt(2), the first five fractions b(n)/a(n) can be read from the following five primitive Pythagorean triples (a(n), b(n), c(n)) = (A195500, A195501, A195502): (3,4,5); |r - b(1)/a(1)| = 0.08... (228,325,397); |r - b(2)/a(2)| = 0.011... (308,435,533); |r - b(3)/a(3)| = 0.0018... (5289,7480,9161); |r - b(4)/a(4)| = 0.000042... (543900,769189,942061); |r - b(5)/a(5)| = 0.0000003...
Shiu := proc(r,n)
t := r+sqrt(1+r^2) ;
cf := numtheory[cfrac](t,n+1) ;
mn := numtheory[nthconver](cf,n) ;
(mn-1/mn)/2 ;
end proc:
A195500 := proc(n)
Shiu(sqrt(2),n) ;
denom(%) ;
end proc: # R. J. Mathar, Sep 21 2011
r = Sqrt[2]; z = 18;
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}]] (* A195500, A195501 *)
Sqrt[a^2 + b^2] (* A195502 *)
r = 2/3; z = 25;
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}]] (* A195559, A195560 *)
Sqrt[a^2 + b^2] (* A195561 *)
(* Peter J. C. Moses, Sep 02 2011 *)
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