1,1

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.

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.

Cf. A003417, A122533.

Sequence in context: A142734 A120773 A116167 this_sequence A080975 A003396 A124899

Adjacent sequences: A122520 A122521 A122522 this_sequence A122524 A122525 A122526

frac,nonn

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