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A072691
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Decimal expansion of Pi^2/12.
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57
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8, 2, 2, 4, 6, 7, 0, 3, 3, 4, 2, 4, 1, 1, 3, 2, 1, 8, 2, 3, 6, 2, 0, 7, 5, 8, 3, 3, 2, 3, 0, 1, 2, 5, 9, 4, 6, 0, 9, 4, 7, 4, 9, 5, 0, 6, 0, 3, 3, 9, 9, 2, 1, 8, 8, 6, 7, 7, 7, 9, 1, 1, 4, 6, 8, 5, 0, 0, 3, 7, 3, 5, 2, 0, 1, 6, 0, 0, 4, 3, 6, 9, 1, 6, 8, 1, 4, 4, 5, 0, 3, 0, 9, 8, 7, 9, 3, 5, 2, 6, 5, 2, 0, 0, 2
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OFFSET
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0,1
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REFERENCES
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C. C. Clawson, The Beauty and Magic of Numbers. New York: Plenum Press (1996): 98
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.11 p. 126 and section 8.5 p. 501.
Jolley, Summation of Series, Dover (1961) eq. (234) page 44.
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LINKS
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Ivan Panchenko, Table of n, a(n) for n = 0..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Brian Hawthorn, The Hardest Integral I've Ever Done, YouTube video, 2021.
Michael Penn, A viewer suggested integral, YouTube video, 2021.
Eric Weisstein's World of Mathematics, Dilogarithm
Index entries for transcendental numbers
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FORMULA
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Equals 1/(1*2) + 1/(2*4) + 1/(3*6) + 1/(4*8) + ... [Jolley]
Equals -dilogarithm(-1). - Rick L. Shepherd, Jul 21 2004
Equals zeta(1,1), the double zeta-function with both arguments equal to 1. - R. J. Mathar, Oct 10 2011
Equals Sum_{n>=1} ((-1)^(n+1))/n^2 [Clawson]. - Alonso del Arte, Aug 15 2012
Equals Integral_{x=0..1} log((1+x^3)/(1-x^3))/x dx. - Bruno Berselli, May 13 2013
From Jean-François Alcover, May 17 2013: (Start)
Equals zeta(2)/2.
Equals Integral_{x=1..2} log(x)/(x-1) dx. (End)
Equals lim_{n->infinity} A244583(n)/prime(n)^2. See A244583 for details. - Richard R. Forberg, Jan 04 2015
Equals Sum_{k>=1} H(k)/(k*2^k), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number. - Amiram Eldar, Aug 20 2020
Equals Integral_{0..infinity} x/(exp(x) + 1) dx. See Abramowitz-Stegun, 23.2.8, for s=2, p. 801. - Wolfdieter Lang, Sep 16 2020
Equals lim_{n->infinity} A024916(n)/(n^2). - Omar E. Pol, Dec 15 2021
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EXAMPLE
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0.822467033424113218236207583323... = A013661/2.
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MATHEMATICA
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RealDigits[Pi^2/12, 10, 105][[1]] (* Robert G. Wilson v *)
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PROG
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(PARI) zeta(2)/2 \\ Michel Marcus, Sep 08 2014
(PARI) -dilog(-1) \\ Charles R Greathouse IV, Apr 17 2015
(PARI) Pi^2/12 \\ Charles R Greathouse IV, Apr 17 2015
(Python)
from mpmath import *
mp.dps=106
print([int(c) for c in list(str(zeta(2)/2))[2:-1]]) # Indranil Ghosh, Jul 08 2017
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CROSSREFS
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Cf. A072692 (Pi^2/12 is in asymptotic formula related to sigma(n), A000203).
Cf. A113319 (sum_{i>=0} 1/(i^2+1)); A232883 (sum_{i>=0} 1/(2*i^2+1)).
Cf. A001008, A002805, A013661, A024916, A237593, A244583.
Sequence in context: A138997 A248498 A133918 * A021928 A185111 A319188
Adjacent sequences: A072688 A072689 A072690 * A072692 A072693 A072694
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KEYWORD
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nonn,cons,changed
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AUTHOR
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Rick L. Shepherd, Jul 02 2002
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STATUS
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approved
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