%I #35 Nov 20 2023 08:18:17
%S 2,4,5,4,3,6,9,2,6,0,6,1,7,0,2,5,9,6,7,5,4,8,9,4,0,1,4,3,1,8,7,1,1,1,
%T 6,2,8,2,7,9,0,3,8,5,9,3,2,6,1,8,0,1,4,2,2,6,3,6,6,7,5,4,6,2,7,4,0,4,
%U 8,1,5,6,7,4,1,1,1,0,0,7,8,0,1,7,8,1,5,2,2,0,7,2,9,8,5,2,8,9,5,9
%N Decimal expansion of Pi/128.
%C Consider 4 circles inscribed in a square. Inscribe a square in each circle. And finally, inscribe 4 circles inside each four small squares. Totally we get 16 small circles. Pi/128 is the ratio of the area of any small circle to the area of the initial square. See the link.
%C Pi/8 (A019675) is the area ratio for the 16 small circles and the initial square.
%C Pi/128 is also the surface area of a sphere whose diameter is 1/sqrt(128). (Cf A222066). - _Omar E. Pol_, May 29 2020
%H Peter Bala, <a href="/A334422/a334422.rtf">A family of series for Pi/128</a>
%H Kirill Ustyantsev, <a href="/A334422/a334422_1.pdf">Geometric interpretation of Pi/128</a>
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F Equals Sum_{k>=1} sin(k)^5*cos(k)^5/k. - _Amiram Eldar_, Jul 11 2020
%F From _Peter Bala_, Nov 17 2023: (Start)
%F Pi/128 = 2*Sum_{k >= 1} k^2/((16*k^2 - 1)*(16*k^2 - 9)).
%F More generally, for n >= 1 we have
%F Pi/128 = (-1)^(n+1) * (2*n)!*Sum_{k >= 1} k^2/( Product_{i = 0..n} 16*k^2 - (2*i + 1)^2 ). (End)
%e 0.02454369260617025967548940143187111628279038593261801422636675462740481567411...
%t RealDigits[Pi/128, 10, 100][[1]] (* _Amiram Eldar_, Apr 30 2020 *)
%o (PARI)
%o default(realprecision, 100);
%o Pi/128
%Y Cf. A000796, A019675, A222066.
%K nonn,cons
%O -1,1
%A _Kirill Ustyantsev_, Apr 30 2020