OFFSET
0,1
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
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.
EXAMPLE
3, 11/4, 295/108, 18839/6912, 2178311/800000, 396789539/145800000, 104534716847/38423222208, ...
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 *)
PROG
(PARI) a(n) = numerator((n+2)^(n+2)/(n+1)^(n+1) - (n+1)^(n+1)/n^n); \\ Michel Marcus, Jun 27 2015
(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
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
N. J. A. Sloane, Oct 17 2005
STATUS
approved