A340738 Denominator of a sequence of fractions converging to e.

1, 2, 7, 18, 71, 252, 1001, 4540, 18089, 99990, 398959, 2602278, 10391023, 78132152, 312129649, 2658297528, 10622799089, 101072656170, 403978495031, 4247085597370, 16977719590391, 195445764537012, 781379079653017, 9775727355457908, 39085931702241241, 528050767520083262, 2111421691000680031

Offset: 1

Author

Gary Detlefs, Jan 18 2021

Keywords

• nonn
• frac

Comments

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).

The convergence is conjectured.

Examples

			Sequence of fractions begins 3/1, 5/2, 19/7, 49/18, 193/71, 685/252, 2721/1001, 12341/4540, ...
		

Crossrefs

Numerators are listed in A340737.

Cf. A007676/A007677.

Programs

Maple

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(1,2,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

Mathematica

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 *)

Formula

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.