A396726 Greatest frequency depth of a word of length n.

1, 3, 4, 5, 6, 6, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11

Offset

1,2

Comments

The frequency depth of a sequence is the number of iterations of the run transform on a sequence required to reach 1. See A327662 for an equivalent definition using different terms.

Note that we only need to consider sequences consisting of 1s and 2s (or any two values, which serve to delineate the start and end of a run).

Links

Table of n, a(n) for n=1..31.

Example

a(5) is 6 because the greatest number of iterations of the run transform before a sequence of length 5 reaches 1 is 6. For example, 1,2,2,1,2 -> 1,2,1,1 -> 1,1,2 -> 2,1 -> 1,1 -> 2 -> 1. This is 6 steps (the most we can get), so a(5) = 6.

Prog

(Python)
from itertools import groupby, product
def RT(t): # run transform of tuple t
    return tuple(len(list(g)) for k, g in groupby(t))
def f(t):
    i = 0
    while t != (1, ): t, i = RT(t), i+1
    return i
def a(n):
    return max(f((2, )+t) for t in product((1, 2), repeat=n-1))
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Jun 03 2026

Crossrefs

Cf. A327662, A362228.

Sequence in context: A133575 A230113 A390458 * A364528 A217031 A104136

Adjacent sequences: A396723 A396724 A396725 * A396727 A396733 A396734

Keyword

nonn,more

Author

Neal Gersh Tolunsky, Jun 03 2026

Extensions

a(22)-a(31) from Michael S. Branicky, Jun 03 2026

Status

approved