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%I #3 Oct 02 2013 16:16:06
%S 57,3667,525153,133794291,53325113593,30632012923107,
%T 23965268215166337,24499823488381227043,31709265214216777648761,
%U 50678828500275334077977523,98023476146668402679417310817
%N Coefficients of the series giving the best rational approximations to 1/e.
%C The series giving the best rational approximations to 1/e is 1/e = 1/3 + 2/a(1) - 2/a(2) + 2/a(3) - ... The continued fraction for 1/e is [0;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 [0;2,1], [0;2,1,2, 1,1], [0;2,1,2,1,1,4,1,1] and so forth are the partial sums of the above infinite series.
%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 A122523, for the best approximations to e.
%Y Cf. A003417, A122523.
%K frac,nonn
%O 1,1
%A _Gene Ward Smith_, Sep 17 2006
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