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A020762
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Decimal expansion of 1/sqrt(5).
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9
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4, 4, 7, 2, 1, 3, 5, 9, 5, 4, 9, 9, 9, 5, 7, 9, 3, 9, 2, 8, 1, 8, 3, 4, 7, 3, 3, 7, 4, 6, 2, 5, 5, 2, 4, 7, 0, 8, 8, 1, 2, 3, 6, 7, 1, 9, 2, 2, 3, 0, 5, 1, 4, 4, 8, 5, 4, 1, 7, 9, 4, 4, 9, 0, 8, 2, 1, 0, 4, 1, 8, 5, 1, 2, 7, 5, 6, 0, 9, 7, 9, 8, 8, 2, 8, 8, 2, 8, 8, 1, 6, 7, 5, 7, 5, 6, 4, 5, 4, 9, 9, 3, 9, 0, 1
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OFFSET
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0,1
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COMMENTS
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This number is the cosine of the central angle of a regular icosahedron; see A105199 for the angle itself. - Clark Kimberling, Feb 10 2009
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LINKS
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Hideyuki Ohtsuka, Problem B-1148, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 52, No. 2 (2014), p. 179; The Exact Value of an Infinite Series, Solution to B-1148, ibid., Vol. 53, No. 2 (2015), pp. 183-184.
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FORMULA
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Equals Sum_{n>=0} (2*n)!/(n!^2*3^(2*n+1)).
Equals Sum_{n>=0} 5*(2*n+1)!/(n!^2*3^(2*n+3)). (End)
Equals Sum_{k>=1} F(2^(k-1))/(L(2^k)+1) = Sum_{k>=0} A058635(k)/(A001566(k)+1), where F(k) = A000045(k) is the k-th Fibonacci number and L(k) = A000032(k) is the k-th Lucas number (Ohtsuka, 2014). - Amiram Eldar, Dec 09 2021
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EXAMPLE
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0.447213595499957939281834733746255247088123671922305144854179449082104...
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MATHEMATICA
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Circs[n_] := With[{r = Sin[Pi/n]/(1 - Sin[Pi/n])}, Graphics[Append[
Table[Circle[(r + 1) {Sin[2 Pi k/n], Cos[2 Pi k/n]}, r], {k, n}],
{Blue, Circle[{0, 0}, 1]}]]]
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PROG
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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