OFFSET
1,1
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 numerators filtered out of this sequence are a(1)..a(8).
The convergence was proved by an autonomous AI agent, see the Lean file. The proof uses the integrals J(k) = Int_{0..1} (x*(1-x))^k * e^x dx, deriving the three-term recurrence J(k+2) = (k+2)*(k+1)*J(k) - 2*(k+2)*(2k+3)*J(k+1) shared by sequences seqU and seqV. Since J(k) tends to 0, the error seqU-seqV*e vanishes, making both even and odd subsequences of the ratio converge to e. - Ralf Stephan, Jun 09 2026
LINKS
Google Deepmind, AlphaProof Nexus: A340737 Lean file.
FORMULA
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.
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(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;
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn,frac,changed
AUTHOR
Gary Detlefs, Jan 18 2021
STATUS
approved