A086463 Decimal expansion of Pi^2/18.

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

Offset

0,1

Comments

The sequence of repeating coefficients [1,-1,-2,-1,1,2] in the sum in the formula section, is equal to the 6th column in A191898. - Mats Granvik, Mar 19 2012

References

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.4.1, p. 20.

A. Holroyd, Sharp Metastability Threshold for Two-Dimensional Bootstrap Percolation, Prob. Th. and Related Fields 125, 195-224, 2003.

Links

Table of n, a(n) for n=0..101.

J. M. Borwein and R. Girgensohn, Evaluations of binomial series, Aequat. Math. 70 (2005) 25-36.

A. Holroyd, Sharp Metastability Threshold for Two-Dimensional Bootstrap Percolation, arXiv:math/0206132 [math.PR], 2002.

Ji-Cai Liu, On two congruences involving Franel numbers, arXiv:2002.03650 [math.NT], 2020.

Courtney Moen, Infinite series with binomial coefficients, Math. Mag. 64 (1) (1991) 53-55.

Renzo Sprugnoli, Sums of reciprocals of the central binomial coefficients, El. J. Combin. Numb. Th. 6 (2006) # A27.

Eric Weisstein's World of Mathematics, Bootstrap Percolation.

Eric Weisstein's World of Mathematics, Central Binomial Coefficient.

Index entries for transcendental numbers.

Formula

Sum[1/n^2/Binomial[2n,n], {n,Infinity}].

Pi^2/18 = A013661/3 = Sum[1/(i+0)^2 - 1/(i+1)^2 - 2/(i+2)^2 - 1/(i+3)^2 + 1/(i+4)^2 + 2/(i+5)^2, {i =1, 7, 13, 19, 25,.. infinity, stride of 6}]. - Mats Granvik, Mar 19 2012

Equals Sum_{k>=1} (H(k) - 2*H(2k))/((-3^k)*k). See Liu. - Michel Marcus, Feb 11 2020

Equals Sum_{k>=1} A007814(k)/k^2. - Amiram Eldar, Jul 13 2020

Equals (2/9) * Sum_{k>=0} (-1)^k*(7*k+5)*k!^3/((2*k+1)*(3*k+2)!) [Gosper 1974] - R. J. Mathar, Feb 07 2024

Continued fraction expansion: 1/(2 - 2/(13 - 48/(34 - 270/(65 - ... - 2*(2*n - 1)*n^3/(5*n^2 + 6*n + 2 - ... ))))). See A130549. - Peter Bala, Feb 16 2024

Example

0.548311355616075478824138388882008396406316633735...

Mathematica

RealDigits[Pi^2/18, 10, 120][[1]] (* Harvey P. Dale, Aug 14 2011 *)

Prog

(PARI) Pi^2/18 \\ Charles R Greathouse IV, Mar 20 2012

Crossrefs

Cf. A007814, A073010, A073016, A086464, A112093, A112094, A130549, A130550.

Sequence in context: A051553 A203139 A184085 * A279916 A021952 A198579

Adjacent sequences: A086460 A086461 A086462 * A086464 A086465 A086466

Keyword

nonn,easy,cons

Author

Eric W. Weisstein, Jul 21 2003

Status

approved