A111130 Numerator of (n+2)^(n+2)/(n+1)^(n+1) - (n+1)^(n+1)/n^n.

3, 11, 295, 18839, 2178311, 396789539, 104534716847, 37582455061871, 17677524703000879, 10535586945520548779, 7758255095720238886679, 6916955444929558486935047, 7342438845112941396534404087, 9150463033951198007724075565619, 13229286823498332297225524829163231

Offset: 0

Author

N. J. A. Sloane, Oct 17 2005

Keywords

• nonn
• frac

Comments

(n+2)^(n+2)/(n+1)^(n+1) - (n+1)^(n+1)/n^n converges very rapidly to e.

These can be prime, as is the case for a(0) = 3, a(1) = 11, a(4) = 18839, a(8) = 37582455061871. These are always odd, just as all but the first denominator of A090205 is even. - Jonathan Vos Post, Oct 19 2005

Examples

			3, 11/4, 295/108, 18839/6912, 2178311/800000, 396789539/145800000, 104534716847/38423222208, ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..210
• H. J. Brothers and J. A. Knox, New closed-form approximations to the logarithmic constant e, Math. Intelligencer, 20 (1998), 25-29.

Crossrefs

Denominators are 1, 4, 108, 6912, ... - see A090205.

Programs

Magma

[Numerator((n+2)^(n+2)/(n+1)^(n+1) - (n+1)^(n+1)/n^n): n in [0..30]]; // G. C. Greubel, Apr 09 2018

Mathematica

Join[{3}, Numerator[Table[(n + 2)^(n + 2)/(n + 1)^(n + 1) - (n + 1)^(n + 1)/n^n, {n, 1, 25}]]] (* G. C. Greubel, Apr 09 2018 *)

PARI

a(n) = numerator((n+2)^(n+2)/(n+1)^(n+1) - (n+1)^(n+1)/n^n); \\ Michel Marcus, Jun 27 2015