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Decimal expansion of Pi/128.
3

%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