A334539 The eventual period of a sequence b(n, m) where b(n, 1) = 1 and the m-th term is the number of occurrences of b(n, m-1) in the list of integers from b(n, max(m-n, 1)) to b(n, m-1).

1, 3, 8, 11, 25, 20, 40, 9, 45, 41, 158, 200, 14, 185, 636, 589, 595, 432, 773, 3196, 1249, 50, 7703, 7661, 12954, 25629, 14885, 41189, 23200, 87410, 33969, 63225, 20486, 212825, 58621, 152952, 135263, 2743830, 729008, 384150, 908629, 126746, 4543899, 3448777, 8531396

Offset: 1

Author

Mark Bedaywi, May 05 2020

Keywords

• nonn

Comments

To generate the sequence b(n, m) for some n, start with the value 1 and then repeatedly append the number of times the last element of the sequence appears in the previous n terms. b(n, m) eventually becomes periodic for all n.

By the pigeonhole principle, a(n) has an upper bound of n^n.

The growth of a(n) appears to be roughly exponential.

Examples

			The sequence b(3, m) is 1, 1, 2, 1, 2, 2, 2, 3, 1, 1, 2, ... the period of which is 8.
The sequence b(4, m) is 1, 1, 2, 1, 3, 1, 2, 1, 2, 2, 3, 1, 1, 2, ... the period of which is 11.
The sequence b(5, m) is 1, 1, 2, 1, 3, 1, 3, 2, 1, 2, 2, 3, 1, 2, 3, 2, 2, 3, 2, 3, 2, 3, 3, 3, 4, 1, 1, 2, ... the period of which is 25.
		

Links

• Elad Michael, Table of n, a(n) for n = 1..100 (terms 1..76 from Johan Westin)
• Reddit user supermac30, Foggy Sequences

Programs

Mathematica

a[k_] := Block[{b = Append[0 Range@k, 1], A=<||>, n=0}, While[True, n++; b = Rest@ b; AppendTo[b, Count[b, b[[-1]]]]; If[ KeyExistsQ[A, b], Break[]]; A[b] = n]; n - A[b]]; Array[a, 30] (* Giovanni Resta, May 06 2020 *)

Python

import sympy
def A334539(n):
  return next(sympy.cycle_length(lambda x:x[1:]+(x.count(x[-1]),),(0,)*(n-1)+(1,)))[0] # Pontus von Brömssen, May 05 2021

Extensions

More terms from Giovanni Resta, May 06 2020