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

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, 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, 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

Offset: 0

Author

Omar E. Pol, May 20 2022

Keywords

• nonn
• cons
• easy

Comments

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).

Examples

			0.177532966575886781763792416676987405390525049396600781132220885314996264798...
		

References

• Ovidiu Furdui, Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis, New York: Springer, 2013.

Links

• Ovidiu Furdui, Problem A, Nieuw Archief voor Wiskunde, Vol. 9, No. 1 (2008), p. 86; "Problem 2008/1-A, Solution to Problem A by Noud Aldenhoven and Daan Wanrooy, ibid., Vol. 9, No. 3 (2008), p. 303.
• Ovidiu Furdui, Problem 1930, Mathematics Magazine, Vol. 86, No. 4 (2013), p. 289; A zeta series, Solution to Problem 1930 by Omran Kouba, ibid., Vol. 87, No. 4 (2014), pp. 296-298.
• Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), (C7.4).
• Index entries for transcendental numbers.

Crossrefs

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

Programs

Mathematica

RealDigits[1 - Pi^2/12, 10, 100][[1]] (* Amiram Eldar, May 20 2022 *)

PARI

1-Pi^2/12

PARI

1-zeta(2)/2

Formula

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

Equals 1 - A013661/2.

Equals 1 - A072691.

Equals A152416/2.

Equals Sum_{k>=1} 1/(2*k*(k+1)^2). - Amiram Eldar, May 20 2022

Equals -1/4 + Sum_{k>=2} (-1)^k * k * (k - Sum_{i=2..k} zeta(i)) (Furdui, 2013 problem). - Amiram Eldar, Jun 09 2022

Equals Integral_{x>=1} {x}/x^3 dx where {.} is the fractional part. [Nahin]. R. J. Mathar, May 22 2024

From Amiram Eldar, Jul 31 2025: (Start)

Equals Integral_{x=0..1} {1/x} * x dx (Furdui, 2013 book, section 2.21, page 103).

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)