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A117759
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Decimal expansion of a certain constant (see Comments lines for definition).
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0
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5, 7, 2, 1, 5, 0, 1, 0, 3, 7, 4, 2, 6, 6, 4, 9, 2, 8, 4, 4, 3, 3, 5, 6, 7, 5, 5, 1, 0, 7, 8, 0, 1, 7, 8, 0, 0, 5, 9, 9, 5, 5, 5, 9, 5, 2, 9, 8, 9, 4, 4, 7, 5, 3, 0, 5, 3, 3, 7, 1, 4, 5, 3, 1, 0, 1, 6, 6, 8, 2, 0, 8, 6, 4, 4, 1, 2, 3, 8, 7, 5, 9, 5, 8, 0, 5, 0, 4, 7, 3, 5, 8, 7, 2, 9, 5, 0, 2, 1, 0, 4
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OFFSET
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1,1
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COMMENTS
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Fanfolding the triangles defined by the spiral of Theodorus, each successive hypotenuse converges to a limit which divides the angle Pi/4 of the first triangle. The ratio of the limit angle to Pi/4 is 0.5721501037... = -sum[((-1)^n)(arctan(1/sqrt(n))] / (Pi/4) The index n starts at 1 for the first triangle (Pi/4) and counts triangles.
I have developed an algorithm to compute the alternating sum, which converges quite slowly. I have computed a value to the precision limit of 64-bit floating point, which is about 14 decimal places. The execution time is about one second. It is unclear whether the algorithm readily adapts to arbitrary precision calculation software in a usable way. There are related sequences: the actual convergent angle in degrees, or radians, etc. as well as a whole family of similarly defined convergents. Other than general treatments of the spiral of Theodorus, no references are known.
Using the sumalt() function in PARI allows one to find as many terms as desired (see PARI program). - Joerg Arndt, Jan 05 2011
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LINKS
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FORMULA
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0.5721501037... = -Sum_{n >= 1} [((-1)^n)(arctan(1/sqrt(n))] / (Pi/4) or = -argument(product[(sqrt(n)+((-1)^n)i)/magnitude(sqrt(n)+((-1)^n)i)]) / (Pi/4).
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MATHEMATICA
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digits = 101; -NSum[(-1)^n*ArcTan[1/Sqrt[n]], {n, 1, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> digits+10]/(Pi/4) // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 15 2013 *)
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PROG
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(PARI): default(realprecision, 155);
-sumalt(n=1, (-1)^n*(atan(1/sqrt(n))))/(Pi/4)
/* gives 0.57215010374266492844335675510780178005995559529894... */
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CROSSREFS
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KEYWORD
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AUTHOR
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Peter Hammer, Apr 14 2006
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EXTENSIONS
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STATUS
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approved
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