A384738 Decimal expansion of 3*log(2)/4 - Pi/8.

1, 2, 7, 1, 6, 1, 3, 0, 3, 7, 2, 1, 2, 3, 4, 8, 2, 7, 2, 5, 5, 0, 9, 3, 6, 6, 8, 1, 8, 3, 6, 9, 4, 5, 6, 5, 5, 3, 1, 9, 7, 8, 9, 2, 5, 8, 4, 8, 3, 0, 3, 2, 1, 2, 9, 6, 8, 6, 4, 1, 9, 3, 3, 0, 8, 1, 5, 6, 8, 1, 6, 5, 6, 9, 1, 4, 9, 4, 9, 1, 1, 8, 7, 5, 8, 9, 3

Offset: 0

Author

Jason Bard, Jun 08 2025

Keywords

• nonn
• cons

Comments

Generalization of infinite sum generating A002162 (natural logarithm of 2). That sum is Sum_{i >= 1} 1/(k*i-1) - 1/(k*i), where k = 2. Here, we set k = 4.

Examples

			0.12716130372123482725509366818369456553197892584830...
		

Links

• Jason Bard, Table of n, a(n) for n = 0..9999
• Steve Chow (Blackpenredpen), Not telescoping: series of 1/(4n-1)-1/(4n) (2019), YouTube video.
• Michael I. Shamos, A catalog of the real numbers, (2007). See p. 166.

Crossrefs

Cf. A002162, A100046, A384683.

Programs

Mathematica

RealDigits[3/4*Log[2] - Pi/8, 10, 140][[1]]

PARI

3*log(2)/4 - Pi/8 \\ Amiram Eldar, Jun 09 2025

Formula

Equals Sum_{k>=1} (1/(4k-1) - (1/4k)).

Equals Sum_{k>=2} zeta(k)/4^k.

Equals Integral_{x=1..oo} 1/(x^4+x^3+x^2+x) dx.

Equals A100046/2. - Amiram Eldar, Jun 09 2025