A374723 - OEIS

%I #55 Sep 05 2024 03:29:09

%S 1,4,120,74455920,479301523749226879680,

%T 1336367227479573478314645756081634359006026455038720

%N Numerator of rational number tan(2^(n-1)*arctan(1/A024810(n))).

%C 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.

%H Sanjar M. Abrarov, Rajinder K. Jagpal, Rehan Siddiqui, and Brendan M. Quine, <a href="https://doi.org/10.3390/mca29020017">An Iterative Method for Computing π by Argument Reduction of the Tangent Function</a>, Math. Comput. Appl. 2024, 29(2), 17.

%F a(n) = numerator(tan(2^(n-1)*arctan(1/A024810(n)))).

%t a[0] := 0;

%t a[n_] := Sqrt[2 + a[n - 1]];

%t b[n_] := Floor[a[n]/Sqrt[2 - a[n - 1]]];

%t r[0, x_] := 1;

%t r[1, x_] := (2*x)/(1 - x^2);

%t r[n_, x_] := (2*r[n - 1, x])/(1 - r[n - 1, x]^2);

%t k = 1;

%t While[k <= 6, Print[k, " ", Numerator[r[k - 1, 1/b[k]]]]; k++];

%Y Cf. A024810, A375935 (denominators).

%K nonn,frac

%O 1,2

%A _Sanjar Abrarov_, Sep 03 2024