OFFSET
0,3
COMMENTS
The frequencies of the terms follow the Poisson distribution with parameter value 1.
The geometric mean approaches A385686 in the limit. In general, for parameter value p it approaches Product_{k>=2} k^(((p^(k-1))*((e-1)^(-p)))/k!).
This sequence can be transformed to A382093 by replacing every occurrence of 1 with 1,2 and adding a 1 to the other terms. So [1,1,2] -> [1,2,1,2,3]. - Jwalin Bhatt, Mar 13 2026
This sequence is generated by the D'Hondt (or Jefferson) apportionment method for the Poisson distribution conditioned on being positive (choosing the smallest element in case of ties). - Pontus von Brömssen, Mar 30 2026
LINKS
Jwalin Bhatt, Table of n, a(n) for n = 0..9999
Wikipedia, D'Hondt method.
Wikipedia, Poisson distribution.
EXAMPLE
Every 1 is followed by a 1 because 1! = 1,
after every (2!=2) ones we see a 2,
after every (3!=6) ones we see a 3 and so on.
MATHEMATICA
A385685[n_] := Module[{N1 = 0, NR= 1, result = {}, i=1}, While[Length[result] < n, N1++; AppendTo[result, 1]; Do[If[Mod[N1, Factorial[k]] == 0, AppendTo[result, k]; f[k == NR + 1, NR++]], {k, 2, NR + 1}]; If[Length[result] > n, result = Take[result, n]]]; result]; A385685[92] (* James C. McMahon, Jul 11 2025 *)
PROG
(Python)
from itertools import islice
from math import factorial
def poisson_distribution_generator():
num_ones, num_reached = 0, 1
while num_ones := num_ones+1:
yield 1
for num in range(2, num_reached+2):
if num_ones % factorial(num) == 0:
yield num
num_reached += num == num_reached+1
A385685 = list(islice(poisson_distribution_generator(), 120))
CROSSREFS
KEYWORD
nonn
AUTHOR
Jwalin Bhatt, Jul 06 2025
STATUS
approved