A122533 Coefficients of the series giving the best rational approximations to 1/e.

57, 3667, 525153, 133794291, 53325113593, 30632012923107, 23965268215166337, 24499823488381227043, 31709265214216777648761, 50678828500275334077977523, 98023476146668402679417310817

Offset: 1

Author

Gene Ward Smith, Sep 17 2006

Keywords

• frac
• nonn

Comments

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.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A003417, A122523.

Programs

Magma

I:=[57, 3667, 525153]; [n le 3 select I[n] else ((2*n-3)*(16*n^2 - 3)*Self(n-1) + (2*n+1)*(16*(n-1)^2 -3)*Self(n-2) - (2*n+1)*Self(n-3))/(2*n-3): n in [1..30]]; // G. C. Greubel, Oct 27 2024

Mathematica

RecurrenceTable[{a[n]== ((2*n-3)*(16*n^2 -3)*a[n-1] +(2*n+1)*(16*(n-1)^2 - 3)*a[n-2] -(2*n+1)*a[n-3])/(2*n-3), a[1]==57, a[2]==3667, a[3]==525153}, a, {n,30}] (* G. C. Greubel, Oct 27 2024 *)

SageMath

@CachedFunction
def a(n): # a = A122533
    if n<4: return (0,57,3667,525153)[n]
    else: return ((2*n-3)*(16*n^2 -3)*a(n-1) +(2*n+1)*(16*(n-1)^2 -3)*a(n-2) -(2*n+1)*a(n-3))/(2*n-3)
[a(n) for n in range(1,31)] # G. C. Greubel, Oct 27 2024

Formula

a(n+3) = (16*n^2 + 96*n + 141) * a(n+2) + (2*n+7)*(16*n^2 + 64*n + 61)/(2*n+3) * a(n+1) - (2*n+7)/(2*n+3) * a(n). This recurrence relationship is identical to A122523, for the best approximations to e.