1,1

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.

Cf. A003417, A122523.

Sequence in context: A011812 A022231 A012058 this_sequence A015259 A012165 A132783

Adjacent sequences: A122530 A122531 A122532 this_sequence A122534 A122535 A122536

frac,nonn

Gene Ward Smith (genewardsmith(AT)gmail.com), Sep 17 2006