OFFSET
1,2
COMMENTS
r(n) = tan(2^(n-1)*arctan(1/A024810(n))) is always a rational number such that its limit lim_{n->oo} r(n)=1.
LINKS
Sanjar M. Abrarov, Rajinder K. Jagpal, Rehan Siddiqui, and Brendan M. Quine, An Iterative Method for Computing π by Argument Reduction of the Tangent Function, Math. Comput. Appl. 2024, 29(2), 17.
FORMULA
a(n) = numerator(tan(2^(n-1)*arctan(1/A024810(n)))).
MATHEMATICA
a[0] := 0;
a[n_] := Sqrt[2 + a[n - 1]];
b[n_] := Floor[a[n]/Sqrt[2 - a[n - 1]]];
r[0, x_] := 1;
r[1, x_] := (2*x)/(1 - x^2);
r[n_, x_] := (2*r[n - 1, x])/(1 - r[n - 1, x]^2);
k = 1;
While[k <= 6, Print[k, " ", Numerator[r[k - 1, 1/b[k]]]]; k++];
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Sanjar Abrarov, Sep 03 2024
STATUS
approved