A122523 Coefficients of series giving the best rational approximations to e.

7, 497, 71071, 18107089, 7216769351, 4145592145057, 3243346361740927, 3315690551047089761, 4291382388990897826759, 6858633609184481948847121, 13266034908146716343647359647, 30540929340877940990799507474097

Offset

1,1

Comments

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.

Links

Table of n, a(n) for n=1..12.

Formula

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.

Crossrefs

Cf. A003417, A122533.

Sequence in context: A116167 A296175 A319831 * A293180 A203472 A219873

Adjacent sequences: A122520 A122521 A122522 * A122524 A122525 A122526

Keyword

frac,nonn

Author

Gene Ward Smith, Sep 17 2006

Status

approved