A196522 Decimal expansion of Pi*(1+sqrt(2))/8.

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

Author

R. J. Mathar, Oct 03 2011

Keywords

• nonn
• cons
• easy

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

Examples

			0.948059448968519935684815...
		

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.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• Index entries for transcendental numbers

Programs

Magma

SetDefaultRealField(RealField(100)); R:= RealField(); Pi(R)*(1+Sqrt(2))/8; // G. C. Greubel, Oct 05 2018

Mathematica

RealDigits[Pi*(1+Sqrt[2])/8,10,120][[1]] (* Harvey P. Dale, May 31 2013 *)

PARI

default(realprecision, 100); Pi*(1+sqrt(2))/8 \\ G. C. Greubel, Oct 05 2018

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