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A334422
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Decimal expansion of Pi/128.
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3
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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, 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, 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
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OFFSET
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-1,1
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COMMENTS
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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
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LINKS
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FORMULA
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Equals Sum_{k>=1} sin(k)^5*cos(k)^5/k. - Amiram Eldar, Jul 11 2020
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)
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EXAMPLE
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0.02454369260617025967548940143187111628279038593261801422636675462740481567411...
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MATHEMATICA
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RealDigits[Pi/128, 10, 100][[1]] (* Amiram Eldar, Apr 30 2020 *)
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PROG
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(PARI)
default(realprecision, 100);
Pi/128
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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