

A340738


Denominator of a sequence of fractions converging to e.


3



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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(n1)/2  a(n2)  (n2)*a(n3)/2 if n is even, 2*a(n1) + n*a(n2) 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, n1)+n*e(a, b, n2) else if n>3 and n mod 2 = 0 then (n+2)*e(a, b, n1)/2 (e(a, b, n2)+(n2)*e(a, b, n3)/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 * A218684 A337614 A343908
Adjacent sequences: A340735 A340736 A340737 * A340739 A340740 A340741


KEYWORD

nonn,frac


AUTHOR

Gary Detlefs, Jan 18 2021


STATUS

approved



