A105534 Decimal expansion of arctan 1/239.
0, 0, 4, 1, 8, 4, 0, 7, 6, 0, 0, 2, 0, 7, 4, 7, 2, 3, 8, 6, 4, 5, 3, 8, 2, 1, 4, 9, 5, 9, 2, 8, 5, 4, 5, 2, 7, 4, 1, 0, 4, 8, 0, 6, 5, 3, 0, 7, 6, 3, 1, 9, 5, 0, 8, 2, 7, 0, 1, 9, 6, 1, 2, 8, 8, 7, 1, 8, 1, 7, 7, 8, 3, 4, 1, 4, 2, 2, 8, 9, 3, 2, 7, 3, 7, 8, 2, 6, 0, 5, 8, 1, 3, 6, 2, 2, 9, 0, 9, 4, 5, 4, 9, 7, 5
Offset: 0
Examples
0.0041840760020747238645382149...
Links
- D. H. Lehmer, On Arccotangent Relations for π, The American Mathematical Monthly, Vol. 45, No. 10 (Dec., 1938), pp. 657-664.
- Eric Weisstein's World of Mathematics, Machin-Like Formulas
- Index entries for transcendental numbers
Programs
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Mathematica
len = 103; n = RealDigits[N[ArcTan[1/239], len]]; PadLeft[First@ n, len + Abs@ Last@ n] (* Michael De Vlieger, Sep 14 2015 *) Join[{0,0},RealDigits[ArcTan[1/239],10,120][[1]]] (* Harvey P. Dale, Apr 29 2016 *)
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PARI
atan(1/239) \\ Michel Marcus, Sep 24 2014
Formula
4*A105532 - arctan(1/239) = Pi/4 (Machin's formula).
arctan(1/239) = Sum_{n >= 1} i/(n*P(n, 239*i)*P(n-1, 239*i)) = 1/239 - 1/40955996 + 1/8773020079176 - 1/1948832181801673304 + 4/1753293766205137615850855 - ..., where i = sqrt(-1) and P(n, x) denotes the n-th Legendre polynomial. - Peter Bala, Mar 21 2024
Comments