OFFSET
-1,1
COMMENTS
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.
Pi/8 (A019675) is the area ratio for the 16 small circles and the initial square.
Pi/128 is also the surface area of a sphere whose diameter is 1/sqrt(128). (Cf A222066). - Omar E. Pol, May 29 2020
LINKS
Peter Bala, A family of series for Pi/128
Kirill Ustyantsev, Geometric interpretation of Pi/128
FORMULA
Equals Sum_{k>=1} sin(k)^5*cos(k)^5/k. - Amiram Eldar, Jul 11 2020
From Peter Bala, Nov 17 2023: (Start)
Pi/128 = 2*Sum_{k >= 1} k^2/((16*k^2 - 1)*(16*k^2 - 9)).
More generally, for n >= 1 we have
Pi/128 = (-1)^(n+1) * (2*n)!*Sum_{k >= 1} k^2/( Product_{i = 0..n} 16*k^2 - (2*i + 1)^2 ). (End)
EXAMPLE
0.02454369260617025967548940143187111628279038593261801422636675462740481567411...
MATHEMATICA
RealDigits[Pi/128, 10, 100][[1]] (* Amiram Eldar, Apr 30 2020 *)
PROG
(PARI)
default(realprecision, 100);
Pi/128
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Kirill Ustyantsev, Apr 30 2020
STATUS
approved