This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A233044 #17 Dec 09 2013 09:29:02
%S A233044 1,1,5,2,65,24,163,60,1957,720,685,252,109601,40320,98641,36288,
%T A233044 9864101,3628800,13563139,4989600,260412269,95800320,8463398743,
%U A233044 3113510400,47395032961,17435658240,888656868019,326918592000
%N A233044 Pairs p, q for those partial sums p/q of the series e = sum_{n>=0} 1/n! that are not convergents to e.
%C A233044 Sondow (2006) conjectured that 2/1 and 8/3 are the only partial sums of the Taylor series for e that are also convergents to the simple continued fraction for e. Sondow and Schalm (2008, 2010) proved partial results toward the conjecture. Berndt, Kim, and Zaharescu (2012) proved it in full.
%D A233044 J. Sondow and K. Schalm, Which partial sums of the Taylor series for e are convergents to e?, (and a link to the primes 2, 5, 13, 37, 463), part I, in Tapas in Experimental Mathematics, T. Amdeberhan and V. H. Moll, eds., Contemp. Math., vol. 457, American Mathematical Society, Providence, RI, 2008, pp. 273-284.
%H A233044 B. Berndt, S. Kim, and A. Zaharescu, <a href="https://math.temple.edu/events/knopp/abstracts.html">Diophantine approximation of the exponential function and Sondow's conjecture</a>, abstract 2012.
%H A233044 J. Sondow, <a href="http://arxiv.org/abs/0704.1282">A geometric proof that e is irrational and a new measure of its irrationality</a>, Amer. Math. Monthly, 113 (2006), 637-641.
%H A233044 J. Sondow and K. Schalm, <a href="http://arxiv.org/abs/0709.0671">Which partial sums of the Taylor series for e are convergents to e?, (and a link to the primes 2, 5, 13, 37, 463), part II</a>, in Gems in Experimental Mathematics, T. Amdeberhan, L. A. Medina, V. H. Moll, eds., Contemp. Math., vol. 517, American Mathematical Society, Providence, RI, 2010, pp. 349-363.
%F A233044 a(2n-1)/a(2n) = A061354(k)/A061355(k) for some k <> 1 and 3.
%F A233044 a(2n-1)/a(2n) <> A007676(k)/A007677(k) for all k.
%e A233044 1/1, 5/2, 65/24, 163/60, 1957/720, 685/252, 109601/40320, 98641/36288, 9864101/3628800, 13563139/4989600, 260412269/95800320, 8463398743/3113510400, 47395032961/17435658240, 888656868019/326918592000
%Y A233044 Cf. A061354, A061355, A007676, A007677.
%K A233044 nonn
%O A233044 1,3
%A A233044 _Jonathan Sondow_, Dec 07 2013