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%I #53 Nov 19 2023 16:01:01
%S 0,8,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,
%T 3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,
%U 3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3
%N Decimal expansion of 1/12.
%C Multiplied by -1, this is zeta(-1) or zeta(-13), with zeta being the Riemann zeta function. Divided by 10, this is zeta(-3). - _Alonso del Arte_, Jan 05 2011
%C Multiplied by 10, this is 5/6, the resistance in ohm between opposite vertices of a cubical network when each edge has a resistance of 1 ohm. - _Michel Marcus_, Sep 02 2015
%C The variance of a continuous uniform distribution U(a,b) is (1/12)*(b-a)^2. - _Jean-François Alcover_, May 19 2016
%C 5/6 is the Schnirelmann density of the sums of three squares and also the asymptotic density of the set of sums of three squares. See Wagstaff. - _Michel Marcus_, Apr 22 2020
%C -1/12 = zeta(-1) is the Ramanujan sum of 1 + 2 + 3 + .... [see facsimile] and was called "one of the most remarkable formulae in science" [Gannon]. - _Peter Luschny_, Jul 17 2020
%D Bruce C. Berndt, Ramanujan's Notebooks: Part 1, Springer-Verlag, 1985, pp. 135-136
%D Terry Gannon, Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics, Cambridge University Press, 2010, p. 140.
%D L. B. W. Jolley, Summation of series, Dover Publications Inc. (New York), 1961, p. 40 (series n. 209) and p. 44 (series n. 239).
%H Martin Gardner, <a href="http://assets.cambridge.org/97805217/56105/excerpt/9780521756105_excerpt.pdf">The Five Platonic Solids</a>, Mathematical Puzzles & Diversions.
%H S. Ramanujan, <a href="https://en.wikipedia.org/wiki/File:Ramanujan_Notebook_1_Chapter_8_on_1234_series.jpg">Another way of finding the constant</a>, Notebook 1, 1919.
%H Samuel S. Wagstaff, Jr., <a href="https://doi.org/10.1090/S0002-9939-1975-0379425-4">The Schnirelmann Density of the Sums of Three Squares</a>, Proc. Amer. Math. Soc. 52 (1975), 1-7.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF">1 + 2 + 3 + 4 + ...</a>.
%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).
%F Equals 1/(1*3*5) + 1/(3*5*7) + 1/(5*7*9) + 1/(7*9*11) + ... = Sum_{i >= 0} 1/((2*i+1)*(2*i+3)*(2*i+5)), see Jolley in References. - _Bruno Berselli_, Mar 21 2014
%F Equals 1/(2*3*4) + 1/(3*4*5) + 1/(4*5*6) + 1/(5*6*7) + ... = Sum_{i > 0} 1/((i+1)*(i+2)*(i+3)). See Jolley in References, p. 48 (sum obtained from the series 268, case t = 2). - _Bruno Berselli_, Mar 29 2014
%F Equals 2*Pi*Integral_{z=-oo..oo} (z/(e^(-Pi*z) + e^(Pi*z)))^2. - _Peter Luschny_, Jul 17 2020
%e 0.083333333333333333333333333333333333333333333333333333333333333333...
%t RealDigits[1/12, 10, 100, -1][[1]] (* _Bruno Berselli_, Mar 21 2014 *)
%o (PARI) 1/12. \\ _Michel Marcus_, Mar 11 2018
%Y Cf. A005408 (odd numbers).
%K nonn,cons,easy
%O 0,2
%A _N. J. A. Sloane_.
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