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Coefficients of the series giving the best rational approximations to 1/e.
2

%I #8 Oct 27 2024 09:26:04

%S 57,3667,525153,133794291,53325113593,30632012923107,

%T 23965268215166337,24499823488381227043,31709265214216777648761,

%U 50678828500275334077977523,98023476146668402679417310817

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

%C 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.

%H G. C. Greubel, <a href="/A122533/b122533.txt">Table of n, a(n) for n = 1..200</a>

%F 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.

%t 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 *)

%o (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

%o (SageMath)

%o @CachedFunction

%o def a(n): # a = A122533

%o if n<4: return (0,57,3667,525153)[n]

%o 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 [a(n) for n in range(1,31)] # _G. C. Greubel_, Oct 27 2024

%Y Cf. A003417, A122523.

%K frac,nonn

%O 1,1

%A _Gene Ward Smith_, Sep 17 2006