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
Links
- G. C. Greubel, Table of n, a(n) for n = 1..200
Programs
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Magma
I:=[7,497,71071]; [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
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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]==7, a[2]==497, a[3]==71071}, a, {n, 30}] (* G. C. Greubel, Oct 27 2024 *) -
SageMath
@CachedFunction def a(n): # a = A122523 if n<4: return (0,7,497,71071)[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 A122533, for the best approximations to 1/e.
Comments