A047872 a(n) = floor(abs(B(2*n + 2)/B(2*n))) where B(n) is the n-th Bernoulli number.

0, 0, 0, 1, 2, 3, 4, 6, 7, 9, 11, 13, 16, 19, 22, 25, 28, 31, 35, 39, 43, 47, 52, 57, 62, 67, 72, 78, 83, 89, 95, 102, 108, 115, 122, 129, 136, 144, 152, 160, 168, 176, 185, 193, 202, 212, 221, 231, 240, 250, 260, 271, 281, 292, 303, 314, 326, 337, 349, 361, 373

Offset

0,5

References

Glaisher, J. W. L.; Tables of the first 250 Bernoulli numbers. Trans. Cambridge Phil. Soc. 12 (1873), 384-391.

Peters, J. and Stein, J., Matematische Tafeln. Revised Russian Edition, 1968, Moscow.

Links

Robert Israel, Table of n, a(n) for n = 0..10000

Maths StackExchange, Bernoulli numbers and Pi^2, 2019.

Index entries for sequences related to Bernoulli numbers.

Formula

a(n) = floor( (n+1)*(2*n+1)/(2*Pi^2)) (conjectured). - Bill McEachen, Dec 08 2021

A002939(n+1)*B(2*n)/B(2*(n+1)) = -(2*Pi)^2*(1 + O(1/4^n)). See the StackExchange link. - Peter Luschny, Dec 08 2021

Example

a(3) = floor(abs(B(4)/B(3))) = floor((1/30)/(1/42)) = floor(7/5) = floor(1.4) = 1.
a(249) = floor(abs(B(250)/B(249))) = 6319.

Maple

seq(floor(abs(bernoulli(2*n+2)/bernoulli(2*n))), n=0..200); # Robert Israel, Jun 27 2018

Mathematica

Table[Floor[Abs[BernoulliB[2*n + 2]/BernoulliB[2*n]]], {n, 0, 60}] (* T. D. Noe, Jun 27 2013 *)

Prog

(PARI) a(n) = floor(abs(bernfrac(2*n+2)/bernfrac(2*n))) \\ Michel Marcus, Jun 27 2013

Crossrefs

Cf. A034972, A034971, A002939.

Sequence in context: A249020 A258470 A248357 * A173254 A015851 A225529

Adjacent sequences: A047869 A047870 A047871 * A047873 A047874 A047875

Keyword

easy,nonn

Author

Labos Elemer

Status

approved