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A340737
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Numerators of a sequence of fractions converging to e.
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3
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3, 5, 19, 49, 193, 685, 2721, 12341, 49171, 271801, 1084483, 7073725, 28245729, 212385209, 848456353, 7226001865, 28875761731, 274743964621, 1098127402131, 11544775603241, 46150226651233, 531276670190245, 2124008553358849, 26573182030311229, 106246577894593683, 1435390805853694145
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OFFSET
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1,1
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COMMENTS
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This sequence is a subset of the numerators of a sequence of fractions converging to e which was obtained by the use of a program which searched for a fraction having a closer value to e than the preceding one. The initial terms of this sequence were 3/1, 5/2, 8/3, 11/4, 19/7, 49/18, 68/25, 87/32, 106/39, 193/71, 685/252, 878/323, 1071/394, 1264/465, 1457/536, 2721/1001, 12341/4540. The subset of the numerators filtered out of this sequence are a(1)..a(8).
The convergence is conjectured.
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LINKS
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FORMULA
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a(1) = 3, a(2) = 5; for n > 2, a(n) = (n+2)*a(n-1)/2 - a(n-2) - (n-2)*a(n-3)/2 if n is even, 2*a(n-1) + n*a(n-2) otherwise.
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EXAMPLE
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Sequence of fractions begins 3/1, 5/2, 19/7, 49/18, 193/71, 685/252, 2721/1001, 12341/4540, ...
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MAPLE
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e:=proc(a, b, n)option remember; e(a, b, 1):=a; e(a, b, 2):=b; if n>2 and n mod 2 =1 then 2*e(a, b, n-1)+n*e(a, b, n-2) else if n>3 and n mod 2 = 0 then (n+2)*e(a, b, n-1)/2 -(e(a, b, n-2)+(n-2)*e(a, b, n-3)/2) fi fi end
seq(e(3, 5, n), n = 1..20)
# code to print the sequence of fractions and error
for n from 1` to 20 do print(e(3, 5, n)/e(1, 2, n), evalf(exp(1)-e(3, 5, n)/e(1, 2, n)) od
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MATHEMATICA
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a[1] = 3; a[2] = 5; a[n_] := a[n] = If[EvenQ[n], (n + 2)*a[n - 1]/2 - (a[n - 2] + (n - 2)*a[n - 3]/2), 2*a[n - 1] + n*a[n - 2]]; Array[a, 20] (* Amiram Eldar, Jan 18 2021 *)
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CROSSREFS
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Denominators are listed in A340738.
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KEYWORD
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nonn,frac
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AUTHOR
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STATUS
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approved
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