A210425 Number of 3-divided words of length n over a 4-letter alphabet.

0, 0, 4, 60, 374, 1960, 9103, 40497, 174127, 735268, 3064477, 12664101, 52005445, 212595280, 866047122

Offset: 1

Author

R. J. Mathar, Mar 21 2012

Keywords

• nonn

Comments

See A210109 for further information.

Row sums of the following table which shows how many words of length n over a 4-letter alphabet are 3-divided in k>=1 different ways:

4;

41, 14, 5;

147, 111, 67, 29, 14, 6;

594, 358, 381, 211, 156, 128, 80, 28, 17, 7;

2072, 1400, 1433, 875, 821, 669, 588, 369, 340, 240, 163, 72, 33, 20, 8;

- R. J. Mathar, Mar 25 2012

Crossrefs

Cf. A210109, A210426.

Programs

Python

from itertools import product
def is3div(b):
    for i in range(1, len(b)-1):
        for j in range(i+1, len(b)):
            X, Y, Z = b[:i], b[i:j], b[j:]
            if all(b < bp for bp in [X+Z+Y, Z+Y+X, Y+X+Z, Y+Z+X, Z+X+Y]):
                return True
    return False
def a(n): return sum(is3div("".join(b)) for b in product("0123", repeat=n))
print([a(n) for n in range(1, 9)]) # Michael S. Branicky, Aug 30 2021

Extensions

a(12)-a(15) from Michael S. Branicky, Aug 30 2021