

A334539


The eventual period of a sequence b(n, m) where b(n, 1) = 1 and the mth term is the number of occurrences of b(n, m1) in the list of integers from b(n, max(mn, 1)) to b(n, m1).


2



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



OFFSET

1,2


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.


LINKS

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


EXAMPLE

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.


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 *)


PROG

(Python)
import sympy
def A334539(n):
return next(sympy.cycle_length(lambda x:x[1:]+(x.count(x[1]), ), (0, )*(n1)+(1, )))[0] # Pontus von BrÃ¶mssen, May 05 2021


CROSSREFS

Sequence in context: A171672 A341262 A070073 * A058565 A170901 A201882
Adjacent sequences: A334536 A334537 A334538 * A334540 A334541 A334542


KEYWORD

nonn,changed


AUTHOR

Mark Bedaywi, May 05 2020


EXTENSIONS

More terms from Giovanni Resta, May 06 2020


STATUS

approved



