This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A122523 #8 Oct 27 2024 09:26:39
%S A122523 7,497,71071,18107089,7216769351,4145592145057,3243346361740927,
%T A122523 3315690551047089761,4291382388990897826759,6858633609184481948847121,
%U A122523 13266034908146716343647359647,30540929340877940990799507474097
%N A122523 Coefficients of series giving the best rational approximations to e.
%C A122523 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.
%H A122523 G. C. Greubel, <a href="/A122523/b122523.txt">Table of n, a(n) for n = 1..200</a>
%F A122523 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.
%t A122523 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 *)
%o A122523 (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
%o A122523 (SageMath)
%o A122523 @CachedFunction
%o A122523 def a(n): # a = A122523
%o A122523 if n<4: return (0,7,497,71071)[n]
%o A122523 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)
%o A122523 [a(n) for n in range(1,31)] # _G. C. Greubel_, Oct 27 2024
%Y A122523 Cf. A003417, A122533.
%K A122523 frac,nonn
%O A122523 1,1
%A A122523 _Gene Ward Smith_, Sep 17 2006