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

%I

%S 3,11,295,18839,2178311,396789539,104534716847,37582455061871,

%T 17677524703000879,10535586945520548779,7758255095720238886679,

%U 6916955444929558486935047,7342438845112941396534404087,9150463033951198007724075565619,13229286823498332297225524829163231

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

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

%C 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

%H G. C. Greubel, <a href="/A111130/b111130.txt">Table of n, a(n) for n = 0..210</a>

%H H. J. Brothers and J. A. Knox, <a href="http://www.brotherstechnology.com/docs/mi_paper1.pdf">New closed-form approximations to the logarithmic constant e</a>, Math. Intelligencer, 20 (1998), 25-29.

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

%t 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 *)

%o (PARI) a(n) = numerator((n+2)^(n+2)/(n+1)^(n+1) - (n+1)^(n+1)/n^n); \\ _Michel Marcus_, Jun 27 2015

%o (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

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

%K nonn,frac

%O 0,1

%A _N. J. A. Sloane_, Oct 17 2005

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Last modified September 24 07:28 EDT 2020. Contains 337317 sequences. (Running on oeis4.)