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 A145008 #15 Oct 22 2013 08:07:39 %S A145008 5,41,3281,21523361,926510094425921,1716841910146256242328924544641, %T A145008 5895092288869291585760436430706259332839105796137920554548481 %N A145008 Reduced numerators of the convergents to 2 = sqrt(4) using the recursion x -> (4/x + x)/2. %C A145008 The recursion x -> (n/x + x)/2 converges to a square root of n. %C A145008 These are the numerators of the first order Newton method to solve x^2-4=f(x)=0, starting at x=1 as the initial estimate: x -> x-f(x)/f'(x), where f'(x)=2x is the first derivative. - _R. J. Mathar_, Oct 07 2008 %C A145008 The equivalent sequence for n=9 starting from x=1 is 5, 17, 257,.., apparently A000215. - _R. J. Mathar_, Oct 14 2008 %H A145008 Wikipedia, <a href="http://en.wikipedia.org/wiki/Newton's_method">Newton's method</a>. %e A145008 (4/1+1)/2 = 5/2 = 2.5 %e A145008 (4/5/2+5/2)/2 = 41/20 = 2.05 %e A145008 (4/(41/20)+41/20)/2 = 3281/1640 = 2.000609... %o A145008 (PARI) g(n,p) = x=1;for(j=1,p,x=(n/x+x)/2;print1(numerator(x)",")) %o A145008 g(4,8) %Y A145008 Cf. A059917. %K A145008 frac,nonn %O A145008 1,1 %A A145008 _Cino Hilliard_, Sep 28 2008 %E A145008 Divided the right hand side of formula in the first comment by 2. - _R. J. Mathar_, Oct 14 2008