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A110255
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Numerators in the fractional coefficients that form the partial quotients of the continued fraction representation of the inverse tangent of 1/x.
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7
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1, 3, 5, 28, 81, 704, 325, 768, 20825, 311296, 83349, 1507328, 1334025, 3145728, 5337189, 130023424, 1366504425, 7516192768, 5466528925, 12884901888, 87470372561, 2954937499648, 349899121845, 12919261626368, 22394407746529
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OFFSET
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1,2
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COMMENTS
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LINKS
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EXAMPLE
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arctan(1/x) = 1/x - 1/(3*x^3) + 1/(5*x^5) - 1/(7*x^7) +-...
= [0; x, 3*x, 5/4*x, 28/9*x, 81/64*x, 704/225*x, 325/256*x,
768/245*x, 20825/16384*x, 311296/99225*x, 83349/65536*x,
1507328/480249*x, 1334025/1048576*x, 3145728/1002001*x,...]
= 1/(x + 1/(3*x + 1/(5/4*x + 1/(28/9*x + 1/(81/64*x +...))))).
The coefficients of x in the even-indexed partial quotients converge to Pi:
{3, 28/9, 704/225, 768/245, 311296/99225, ...}.
The coefficients of x in the odd-indexed partial quotients converge to 4/Pi:
{1, 5/4, 81/64, 325/256, 20825/16384, ...}.
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PROG
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(PARI) {a(n)=numerator(subst((contfrac( sum(k=0, n, (-1)^k/x^(2*k+1)/(2*k+1)), n+1))[n+1], x, 1))}
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CROSSREFS
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KEYWORD
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cofr,frac,nonn
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AUTHOR
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STATUS
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approved
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