cp's OEIS Frontend

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

%I A152416 #47 Aug 06 2025 01:07:37
%S A152416 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,
%T A152416 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,
%U A152416 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
%N A152416 Decimal expansion of 2 - Pi^2/6.
%C A152416 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).
%C A152416 The proportion of triangles formed by random lines in a plane (see Theorem 6 in Miles link). - _Michel Marcus_, Sep 04 2015
%H A152416 R. J. Mathar, <a href="https://arxiv.org/abs/0803.0900">Series of reciprocal powers of k-almost primes</a>, arXiv:0803.0900 (2009) Section 4.1.
%H A152416 Mathematical Reflections, <a href="https://www.awesomemath.org/wp-pdf-files/math-reflections/mr-2013-04/mr_3_2013_solutions.pdf">Solution to Problem U268</a>, Issue 3, 2013, p. 17.
%H A152416 R. E. Miles, <a href="https://doi.org/10.1073/pnas.52.4.901">Random polygons determined by random lines in a plane</a>, PNAS 1964 52 (4) 901-907.
%H A152416 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F A152416 Equals 2 - A013661.
%F A152416 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
%F A152416 From _Amiram Eldar_, Aug 09 2020: (Start)
%F A152416 Equals Sum_{k>=1} 1/(k*(k+1)^2) = Sum_{k>=2} 1/A045991(k).
%F A152416 Equals Integral_{x=0..1} log(x)*log(1-x) dx. (End)
%F A152416 Equals Sum_{i, j >= 1} 1/((i + 1)^2*binomial(i+j+1, i)). - _Peter Bala_, Aug 05 2025
%e A152416 Equals 0.355065933151773563527584833353974810781050098793201562264441770...
%p A152416 evalf(2-Pi^2/6);
%t A152416 First@ RealDigits[N[2 - Pi^2/6, 120]] (* _Michael De Vlieger_, Sep 04 2015 *)
%o A152416 (PARI) 2 - Pi^2/6 \\ _Michel Marcus_, Jan 06 2017
%Y A152416 Cf. A013661, A045991.
%K A152416 cons,easy,nonn
%O A152416 0,1
%A A152416 _R. J. Mathar_, Dec 03 2008