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

1,2

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.

Links

Table of n, a(n) for n=1..27.

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.

Example

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

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 *)

Crossrefs

Numerators are listed in A340737.

Cf. A007676/A007677.

Sequence in context: A185308 A002214 A303742 * A368593 A218684 A337614

Adjacent sequences: A340735 A340736 A340737 * A340739 A340740 A340741

Keyword

nonn,frac

Author

Gary Detlefs, Jan 18 2021

Status

approved