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A354238 Decimal expansion of 1 - Pi^2/12.

%I A354238 #50 Jul 31 2025 08:42:26
%S A354238 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 A354238 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 A354238 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 A354238 Decimal expansion of 1 - Pi^2/12.
%C A354238 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).
%D A354238 Ovidiu Furdui, Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis, New York: Springer, 2013.
%H A354238 Ovidiu Furdui, <a href="https://www.nieuwarchief.nl/serie5/pdf/naw5-2008-09-1-086.pdf">Problem A</a>, Nieuw Archief voor Wiskunde, Vol. 9, No. 1 (2008), p. 86; <a href="https://www.nieuwarchief.nl/serie5/pdf/naw5-2008-09-4-302.pdf">"Problem 2008/1-A</a>, Solution to Problem A by Noud Aldenhoven and Daan Wanrooy, ibid., Vol. 9, No. 3 (2008), p. 303.
%H A354238 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 A354238 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 A354238 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A354238 Equals lim_{n->infinity} A004125(n)/(n^2).
%F A354238 Equals 1 - A013661/2.
%F A354238 Equals 1 - A072691.
%F A354238 Equals A152416/2.
%F A354238 Equals Sum_{k>=1} 1/(2*k*(k+1)^2). - _Amiram Eldar_, May 20 2022
%F A354238 Equals -1/4 + Sum_{k>=2} (-1)^k * k * (k - Sum_{i=2..k} zeta(i)) (Furdui, 2013 problem). - _Amiram Eldar_, Jun 09 2022
%F A354238 Equals Integral_{x>=1} {x}/x^3 dx where {.} is the fractional part. [Nahin]. _R. J. Mathar_, May 22 2024
%F A354238 From _Amiram Eldar_, Jul 31 2025: (Start)
%F A354238 Equals Integral_{x=0..1} {1/x} * x dx (Furdui, 2013 book, section 2.21, page 103).
%F A354238 Equals Integral_{x=0..1} Integral_{y=0..1} {x/y}*{y/x} dx dy, where {} denotes fractional part (Furdui, 2008 and 2013 book, section 2.36, page 105). (End)
%e A354238 0.177532966575886781763792416676987405390525049396600781132220885314996264798...
%t A354238 RealDigits[1 - Pi^2/12, 10, 100][[1]] (* _Amiram Eldar_, May 20 2022 *)
%o A354238 (PARI) 1-Pi^2/12
%o A354238 (PARI) 1-zeta(2)/2
%Y A354238 Cf. A000290, A004125, A013661, A024916, A072691, A152416, A237593, A245092, A353908.
%K A354238 nonn,cons,easy
%O A354238 0,2
%A A354238 _Omar E. Pol_, May 20 2022