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

0,2

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

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

Table of n, a(n) for n=0..98.

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).

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

Example

0.083333333333333333333333333333333333333333333333333333333333333333...

Mathematica

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

Prog

(PARI) 1/12. \\ Michel Marcus, Mar 11 2018

Crossrefs

Cf. A005408 (odd numbers).

Sequence in context: A010148 A246822 A168356 * A269296 A371502 A334363

Adjacent sequences: A021013 A021014 A021015 * A021017 A021018 A021019

Keyword

nonn,cons,easy

Author

N. J. A. Sloane

Status

approved