|
| |
|
|
A123178
|
|
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.
|
|
1
| |
|
|
14, 968, 75920, 6288296, 537005664, 46764723632, 4128230266160, 368090979124960, 33073373083339904, 2989771785328137728, 271603565356722214784, 24774311300942501337728, 2267541753957311770329600
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,1
|
|
|
COMMENTS
| 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).
|
|
|
REFERENCES
| Frits Beukers, A rational approach to Pi, NAW 5/1 nr.4, december 2000, p. 378
|
|
|
EXAMPLE
| K(5) = -3618728790016/2145 + 537005664*Pi so a(5) = 537005664.
|
|
|
MAPLE
| 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
|
|
|
MATHEMATICA
| 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)
|
|
|
CROSSREFS
| Sequence in context: A171183 A064729 A189304 * A160011 A104226 A132504
Adjacent sequences: A123175 A123176 A123177 * A123179 A123180 A123181
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| Benoit Cloitre (abmt(AT)orange.fr), Oct 03 2006
|
|
|
EXTENSIONS
| More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 07 2006
|
| |
|
|
