A021016 Decimal expansion of 1/12.

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, 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, 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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cons
• easy

Comments

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

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

The variance of a continuous uniform distribution U(a,b) is (1/12)*(b-a)^2. - Jean-François Alcover, May 19 2016

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

-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

Examples

			0.083333333333333333333333333333333333333333333333333333333333333333...
		

References

• Bruce C. Berndt, Ramanujan's Notebooks: Part 1, Springer-Verlag, 1985, pp. 135-136
• Terry Gannon, Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics, Cambridge University Press, 2010, p. 140.
• L. B. W. Jolley, Summation of series, Dover Publications Inc. (New York), 1961, p. 40 (series n. 209) and p. 44 (series n. 239).

Links

• Martin Gardner, The Five Platonic Solids, Mathematical Puzzles & Diversions.
• Srinivasa Ramanujan, Question 463, Journal of the Indian Mathematical Society, Vol. 5 (1913), p. 120.
• Srinivasa Ramanujan, Another way of finding the constant, Notebook 1, 1919.
• Samuel S. Wagstaff, Jr., The Schnirelmann Density of the Sums of Three Squares, Proc. Amer. Math. Soc. 52 (1975), 1-7.
• Wikipedia, 1 + 2 + 3 + 4 + ....
• Index entries for linear recurrences with constant coefficients, signature (1).

Crossrefs

Cf. A005408 (odd numbers), A010701.

Programs

Mathematica

RealDigits[1/12, 10, 100, -1][[1]] (* Bruno Berselli, Mar 21 2014 *)

PARI

1/12. \\ Michel Marcus, Mar 11 2018

Formula

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

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

Equals 2*Pi*Integral_{z=-oo..oo} (z/(e^(-Pi*z) + e^(Pi*z)))^2. - Peter Luschny, Jul 17 2020

Equals lim_{x->oo} (P(x) - (1 - t(x))/(1 + t(x)))^(1/x) = lim_{x->oo} (t(x) - (1 - P(x))/(1 + P(x)))^(1/x) by the inversion, where P(x) is the prime zeta function of x and t(x) = zeta(2x)/zeta(x)^2, with zeta(x) being the Riemann zeta function of x. - Thomas Ordowski, Oct 28 2024

Equals Integral_{x>=0} 1/(exp(2*Pi*sqrt(x))-1) dx (Ramanujan, 1913). - Amiram Eldar, Jan 01 2025

Equals Integral_{x=0..1} x^(1/5) - x^(1/3) dx. - Kritsada Moomuang, May 27 2025