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 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). 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, )*(n-1)+(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

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Last modified May 7 17:40 EDT 2021. Contains 343652 sequences. (Running on oeis4.)