A152416 Decimal expansion of 2 - Pi^2/6.

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

Offset: 0

Author

R. J. Mathar, Dec 03 2008

Keywords

• cons
• easy
• nonn

Comments

Essentially the 9's complement of the digits of A013661, starting with the second. Consider the constants N(s) = Sum_{n >= 2} 1/(n^s*(n-1)) = s - Sum_{k=2..s} Zeta(k), where Zeta is Riemann's zeta function. N(1)=1 and this constant here is N(2).

The proportion of triangles formed by random lines in a plane (see Theorem 6 in Miles link). - Michel Marcus, Sep 04 2015

Examples

			Equals 0.355065933151773563527584833353974810781050098793201562264441770...
		

Links

• R. J. Mathar, Series of reciprocal powers of k-almost primes, arXiv:0803.0900 (2009) Section 4.1.
• Mathematical Reflections, Solution to Problem U268, Issue 3, 2013, p. 17.
• R. E. Miles, Random polygons determined by random lines in a plane, PNAS 1964 52 (4) 901-907.
• Index entries for transcendental numbers

Crossrefs

Cf. A013661, A045991.

Programs

Maple

evalf(2-Pi^2/6);

Mathematica

First@ RealDigits[N[2 - Pi^2/6, 120]] (* Michael De Vlieger, Sep 04 2015 *)

PARI

2 - Pi^2/6 \\ Michel Marcus, Jan 06 2017

Formula

Equals 2 - A013661.

Equals lim_{n->oo} (1/n^2)*Sum_{k=2..n^2-1} (fractional_part(n/sqrt(k))). See Mathematical Reflections link. - Michel Marcus, Jan 06 2017

From Amiram Eldar, Aug 09 2020: (Start)

Equals Sum_{k>=1} 1/(k*(k+1)^2) = Sum_{k>=2} 1/A045991(k).

Equals Integral_{x=0..1} log(x)*log(1-x) dx. (End)

Equals Sum_{i, j >= 1} 1/((i + 1)^2*binomial(i+j+1, i)). - Peter Bala, Aug 05 2025