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

Author

Labos Elemer

Keywords

• easy
• nonn

Examples

			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.
		

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.

Crossrefs

Cf. A034972, A034971, A002939.

Programs

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 *)

PARI

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

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