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Search: id:A123178
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| A123178 |
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K(n)=a(n)*Pi-b(n)/c(n) where K(n)=integral(t=-1,1,t^(2n)*(1-t^2)^(2n)/(1+it)^(3n+1)dt) and a(n),b(n),c(n) are positive integers. |
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+0
1
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| 14, 968, 75920, 6288296, 537005664, 46764723632, 4128230266160, 368090979124960, 33073373083339904, 2989771785328137728, 271603565356722214784, 24774311300942501337728, 2267541753957311770329600
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Integrals K(n) give us a sequence of approximation of Pi whose qualities exceed 1.0449 in the long run. a(n) is divisible by 2^floor(n/2).
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REFERENCES
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Frits Beukers, A rational approach to Pi, NAW 5/1 nr.4, december 2000, p. 378
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EXAMPLE
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K(5) = -3618728790016/2145 + 537005664*Pi so a(5) = 537005664.
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MAPLE
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Kn := proc(n) local a, l ; a := 0 : for l from 0 to (3*n+1)/2 do a := a+2*binomial(3*n+1, 2*l)*(-1)^l* int(t^(2*n+2*l)*(1-t^2)^(2*n)/(1+t^2)^(3*n+1), t=0..1) ; od ; a := subs(Pi=x, a) ; RETURN(a) ; end: A123178 := proc(n) RETURN( coeftayl(Kn(n), x=0, 1)) ; end: for n from 1 to 20 do printf("%d, ", A123178(n)) ; od ; - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 07 2006
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MATHEMATICA
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f[n_] := CoefficientList[ Integrate[t^(2n)*(1 - t^2)^(2n)/(1 + I*t)^(3n + 1), {t, -1, 1}], Pi][[ -1]]; Array[f, 13] (Robert G. Wilson v)
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CROSSREFS
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Sequence in context: A115458 A055152 A064729 this_sequence A160011 A104226 A132504
Adjacent sequences: A123175 A123176 A123177 this_sequence A123179 A123180 A123181
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KEYWORD
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nonn
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AUTHOR
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Benoit Cloitre (abmt(AT)orange.fr), Oct 03 2006
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EXTENSIONS
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More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 07 2006
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