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%I #23 Feb 13 2021 15:07:15
%S 1,2,7,18,71,252,1001,4540,18089,99990,398959,2602278,10391023,
%T 78132152,312129649,2658297528,10622799089,101072656170,403978495031,
%U 4247085597370,16977719590391,195445764537012,781379079653017,9775727355457908,39085931702241241,528050767520083262,2111421691000680031
%N Denominator of a sequence of fractions converging to e.
%C 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 denominators filtered out of this sequence are a(1)..a(8).
%C The convergence is conjectured.
%F a(1) = 1, a(2) = 2; 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.
%e Sequence of fractions begins 3/1, 5/2, 19/7, 49/18, 193/71, 685/252, 2721/1001, 12341/4540, ...
%p 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
%p seq(e(1,2,n), n = 1..20)
%p # code to print the sequence of fractions and error
%p 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
%t a[1] = 1; a[2] = 2; 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 *)
%Y Numerators are listed in A340737.
%Y Cf. A007676/A007677.
%K nonn,frac
%O 1,2
%A _Gary Detlefs_, Jan 18 2021
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