A354238 - OEIS

%I #42 May 22 2024 12:56:38

%S 1,7,7,5,3,2,9,6,6,5,7,5,8,8,6,7,8,1,7,6,3,7,9,2,4,1,6,6,7,6,9,8,7,4,

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

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

%N Decimal expansion of 1 - Pi^2/12.

%C Ratio of area between the polygon that is adjacent in the same plane to the base of the stepped pyramid with an infinite number of levels described in A245092 and the circumscribed square (see the first formula).

%H Ovidiu Furdui, <a href="http://www.jstor.org/stable/10.4169/math.mag.86.4.288">Problem 1930</a>, Mathematics Magazine, Vol. 86, No. 4 (2013), p. 289; <a href="http://www.jstor.org/stable/10.4169/math.mag.87.4.292">A zeta series</a>, Solution to Problem 1930 by Omran Kouba, ibid., Vol. 87, No. 4 (2014), pp. 296-298.

%H Paul J. Nahin, <a href="https://doi.org/10.1007/978-3-030-43788-6">Inside interesting integrals</a>, Undergrad. Lecture Notes in Physics, Springer (2020), (C7.4)

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.

%F Equals lim_{n->infinity} A004125(n)/(n^2).

%F Equals 1 - A013661/2.

%F Equals 1 - A072691.

%F Equals A152416/2.

%F Equals Sum_{k>=1} 1/(2*k*(k+1)^2). - _Amiram Eldar_, May 20 2022

%F Equals -1/4 + Sum_{k>=2} (-1)^k * k * (k - Sum_{i=2..k} zeta(i)) (Furdui, 2013). - _Amiram Eldar_, Jun 09 2022

%F Equals integral_1^oo {x}/x^3 dx where {.} is the fractional part. [Nahin]. _R. J. Mathar_, May 22 2024

%e 0.177532966575886781763792416676987405390525049396600781132220885314996264798...

%t RealDigits[1 - Pi^2/12, 10, 100][[1]] (* _Amiram Eldar_, May 20 2022 *)

%o (PARI) 1-Pi^2/12

%o (PARI) 1-zeta(2)/2

%Y Cf. A000290, A004125, A013661, A024916, A072691, A152416, A237593, A245092, A353908.

%K nonn,cons,easy

%O 0,2

%A _Omar E. Pol_, May 20 2022