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A196522
Decimal expansion of Pi*(1+sqrt(2))/8.
1
9, 4, 8, 0, 5, 9, 4, 4, 8, 9, 6, 8, 5, 1, 9, 9, 3, 5, 6, 8, 4, 8, 1, 5, 5, 4, 6, 6, 6, 7, 5, 2, 4, 5, 7, 2, 8, 5, 1, 4, 7, 3, 8, 8, 6, 0, 9, 3, 8, 4, 9, 5, 0, 5, 5, 0, 7, 5, 4, 2, 5, 2, 4, 9, 0, 8, 0, 3, 1, 3, 9, 9, 9, 2, 3, 2, 1, 0, 3, 6, 3, 6, 5, 0, 4, 2, 0, 2, 2, 0, 0, 1, 3, 3, 6, 0, 2, 8
OFFSET
0,1
COMMENTS
This is the mean of two Dirichlet L=functions modulo m=8 at s=1, one with character (1,0,-1,0,1,0,-1,0) as in A101455, the other with character (1,0,1,0,-1,0,-1,0).
The area of a circle circumscribed in a unit-area regular octagon. - Amiram Eldar, Nov 05 2020
REFERENCES
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.4.1, p. 20.
L. B. W. Jolley, Summation of series, Dover (1961), eq. 78 page 16 and eq. 264 page 48.
FORMULA
Equals (1 - 1/7) + (1/9 - 1/15) + ... + (1/(1+8*k) - 1/(7+8*k)) + ... = (A093954 + A003881)/2.
Equals Sum_{n >= 0} (8*k + 6)/((8*n + 1)*(8*n + 8*k + 7)) - Sum_{n = 0..k-1} 1/(8*n + 7), for positive integer k. - Peter Bala, Jul 10 2024
EXAMPLE
0.948059448968519935684815...
MATHEMATICA
RealDigits[Pi*(1+Sqrt[2])/8, 10, 120][[1]] (* Harvey P. Dale, May 31 2013 *)
PROG
(PARI) default(realprecision, 100); Pi*(1+sqrt(2))/8 \\ G. C. Greubel, Oct 05 2018
(Magma) SetDefaultRealField(RealField(100)); R:= RealField(); Pi(R)*(1+Sqrt(2))/8; // G. C. Greubel, Oct 05 2018
CROSSREFS
Sequence in context: A199665 A113273 A011460 * A334114 A268315 A019659
KEYWORD
nonn,cons,easy,changed
AUTHOR
R. J. Mathar, Oct 03 2011
STATUS
approved