A352770 Decimal expansion of Pi^2/12 - log(2)^2.

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

Offset

0,1

References

Ovidiu Furdui, Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis, New York: Springer, 2013. See Problem 3.64, pages 149 and 213-214.

Links

Table of n, a(n) for n=0..104.

Ovidiu Furdui, Problem 3301, Crux Mathematicorum, Vol. 34, No. 1 (2008), pp. 44 and 46; Solution to Problem 3301 by Manuel Benito, ibid., Vol. 35, No. 1 (2009), pp. 47-49.

MathOverflow, An infinite series involving harmonic numbers, 2021.

Michael Ian Shamos, Shamos's Catalog of the Real Numbers, 2011, p. 383.

Formula

Formulas from Furdui (2008):

Equals Sum_{n>=1} (log(2) - H(2*n) + H(n))/n, where H(n) = A001008(n)/A002805(n) is the n-th harmonic number.

Equals Sum_{n>=1} H(n)/((2*n+1)*(2*n+2)).

Formulas from Shamos (2011):

Equals Sum_{n>=1} H(n)/(2^n*n*(n+1)).

Equals Sum_{n>=1} (-1)^(n+1)*H(n)/(n*(n+1)).

Example

0.34201401950591179356910505699634762287892199900885...

Mathematica

RealDigits[Pi^2/12 - Log[2]^2, 10, 100][[1]]

Prog

(PARI) Pi^2/12 - log(2)^2 \\ Michel Marcus, Apr 02 2022

Crossrefs

Cf. A001008, A002805, A072691, A076788, A253191.

Sequence in context: A241833 A332096 A077451 * A019829 A200125 A091528

Adjacent sequences: A352767 A352768 A352769 * A352771 A352772 A352773

Keyword

nonn,cons,changed

Author

Amiram Eldar, Apr 02 2022

Status

approved