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%I #3 Oct 02 2013 16:16:06
%S 7,497,71071,18107089,7216769351,4145592145057,3243346361740927,
%T 3315690551047089761,4291382388990897826759,6858633609184481948847121,
%U 13266034908146716343647359647,30540929340877940990799507474097
%N Coefficients of series giving the best rational approximations to e.
%C The series giving the best rational approximations to e is e = 3 - 2/a(1) + 2/a(2) - 2/a(3) + ... The continued fraction for e is [2;1,2,1,1,4,1,1,6, 1,1,8...] and the above best approximations give every third convergent, the convergents deriving from [2;1], [2;1,2,1,1], [2;1,2,1,1,4,1, 1] and so forth.
%F a(n+3) = (16*n^2+96*n+141) * a(n+2) + (2*n+7)*(16*n^2+64*n+61)/(2*n+2) * a(n+1) - (2*n+7)/(2*n+3) * a(n). This recurrence relationship is identical to A122533, for the best approximations to 1/e.
%Y Cf. A003417, A122533.
%K frac,nonn
%O 1,1
%A _Gene Ward Smith_, Sep 17 2006
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