The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.
Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations
(which often say "discovered thanks to the OEIS").
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.
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.