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A122523 Coefficients of series giving the best rational approximations to e. 1
7, 497, 71071, 18107089, 7216769351, 4145592145057, 3243346361740927, 3315690551047089761, 4291382388990897826759, 6858633609184481948847121, 13266034908146716343647359647, 30540929340877940990799507474097 (list; graph; refs; listen; history; text; internal format)
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

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Last modified May 15 01:35 EDT 2021. Contains 343909 sequences. (Running on oeis4.)