A356511 Total number of distinct numbers that can be obtained by starting with 1 and applying the "Choix de Bruxelles", version 2 operation at most n times in duodecimal (base 12).

1, 2, 3, 4, 5, 9, 19, 45, 107, 275, 778, 2581, 10170, 45237, 222859, 1191214, 6887258, 42894933, 287397837

Offset: 0

Author

J. Conrad, Aug 09 2022

Keywords

• nonn
• more
• base

Examples

			For n=4, the a(4) = 5 numbers obtained are (in base 12): 1, 2, 4, 8, 14.
For n=5, they expand to a(5) = 9 numbers (in base 12): 1, 2, 4, 8, 12, 14, 18, 24, 28.
		

Links

• Eric Angelini, Lars Blomberg, Charlie Neder, Remy Sigrist, and N. J. A. Sloane, "Choix de Bruxelles": A New Operation on Positive Integers, arXiv:1902.01444 [math.NT], Feb 2019; Fib. Quart. 57:3 (2019), 195-200.
• Eric Angelini, Lars Blomberg, Charlie Neder, Remy Sigrist, and N. J. A. Sloane,, "Choix de Bruxelles": A New Operation on Positive Integers, Local copy.
• J. Conrad, Python program.

Crossrefs

Cf. A323289 (decimal).

Programs

Python

# See Conrad link.

Python

from itertools import islice
from sympy.ntheory import digits
def fd12(d): return sum(12**i*di for i, di in enumerate(d[::-1]))
def cdb2(n):
    d, out = digits(n, 12)[1:], {n}
    for l in range(1, len(d)+1):
        for i in range(len(d)+1-l):
            if d[i] == 0: continue
            t = fd12(d[i:i+l])
            out.add(fd12(d[:i] + digits(2*t, 12)[1:] + d[i+l:]))
            if t&1 == 0:
                out.add(fd12(d[:i] + digits(t//2, 12)[1:] + d[i+l:]))
    return out
def agen():
    reach, expand = {1}, [1]
    while True:
        yield len(reach)
        newreach = {r for q in expand for r in cdb2(q) if r not in reach}
        reach |= newreach
        expand = list(newreach)
print(list(islice(agen(), 14))) # Michael S. Branicky, Aug 17 2022

Extensions

a(16)-a(18) from Michael S. Branicky, Aug 17 2022