Search: id:a257945
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A257945
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Decimal expansion of abs(i/(i + i/(i + i/...))) and abs(i/(1 + i/(1 + i/...))), i being the imaginary unit.
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+0
1
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6, 9, 3, 2, 0, 5, 4, 6, 4, 6, 2, 3, 7, 9, 7, 3, 2, 0, 4, 3, 4, 3, 6, 3, 7, 0, 4, 2, 2, 4, 1, 3, 8, 6, 8, 7, 9, 4, 1, 0, 2, 1, 7, 5, 0, 1, 6, 9, 2, 1, 9, 0, 1, 3, 3, 9, 9, 5, 5, 5, 8, 6, 7, 5, 2, 9, 5, 5, 8, 1, 4, 8, 8, 3, 1, 6, 6, 1, 0, 4, 3, 0, 2, 2, 3, 3, 6, 0, 6, 9, 1, 5, 2, 6, 8, 1, 8, 5, 8, 3, 5, 0, 5, 6, 4
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OFFSET
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0,1
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COMMENTS
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Set v = A156590 and u = (A156548 - 1). Then the continued fractions evaluate to i/(i + i/(i + i/...)) = (sqrt(4*i - 1) - i)/2 = v + u*i and i/(1 + i/(1 + i/...)) = (sqrt(4*i + 1) - 1)/2 = u + v*i. They can be evaluated either explicitly or as limits of the convergent recursive mappings z -> i/(i + z) and z -> i/(1 + z), respectively, starting, for example, with z = 0.
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LINKS
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FORMULA
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Equals sqrt(1 + sqrt(17) - sqrt(2*(1 + sqrt(17))))/2.
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EXAMPLE
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0.69320546462379732043436370422413868794102175016921901339955586752...
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MATHEMATICA
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RealDigits[Sqrt[1 + Sqrt[17] - Sqrt[2*(1 + Sqrt[17])]]/2, 10, 105][[1]] (* Vaclav Kotesovec, Jun 02 2015 *)
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PROG
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(PARI) sqrt(1+sqrt(17)-sqrt(2*(1+sqrt(17))))/2
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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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