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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

%I #30 Jan 09 2025 13:04:15

%S 1,2,3,4,5,9,19,45,107,275,778,2581,10170,45237,222859,1191214,

%T 6887258,42894933,287397837

%N 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).

%H Eric Angelini, Lars Blomberg, Charlie Neder, Remy Sigrist, and N. J. A. Sloane, <a href="http://arxiv.org/abs/1902.01444">"Choix de Bruxelles": A New Operation on Positive Integers</a>, arXiv:1902.01444 [math.NT], Feb 2019; Fib. Quart. 57:3 (2019), 195-200.

%H Eric Angelini, Lars Blomberg, Charlie Neder, Remy Sigrist, and N. J. A. Sloane,, <a href="/A307635/a307635.pdf">"Choix de Bruxelles": A New Operation on Positive Integers</a>, Local copy.

%H J. Conrad, <a href="https://raw.githubusercontent.com/cxr00/cxr/master/tests/base64/choix_de_bruxelles.py">Python program</a>.

%e For n=4, the a(4) = 5 numbers obtained are (in base 12): 1, 2, 4, 8, 14.

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

%o (Python) # See Conrad link.

%o (Python)

%o from itertools import islice

%o from sympy.ntheory import digits

%o def fd12(d): return sum(12**i*di for i, di in enumerate(d[::-1]))

%o def cdb2(n):

%o d, out = digits(n, 12)[1:], {n}

%o for l in range(1, len(d)+1):

%o for i in range(len(d)+1-l):

%o if d[i] == 0: continue

%o t = fd12(d[i:i+l])

%o out.add(fd12(d[:i] + digits(2*t, 12)[1:] + d[i+l:]))

%o if t&1 == 0:

%o out.add(fd12(d[:i] + digits(t//2, 12)[1:] + d[i+l:]))

%o return out

%o def agen():

%o reach, expand = {1}, [1]

%o while True:

%o yield len(reach)

%o newreach = {r for q in expand for r in cdb2(q) if r not in reach}

%o reach |= newreach

%o expand = list(newreach)

%o print(list(islice(agen(), 14))) # _Michael S. Branicky_, Aug 17 2022

%Y Cf. A323289 (decimal).

%K nonn,more,base

%O 0,2

%A _J. Conrad_, Aug 09 2022

%E a(16)-a(18) from _Michael S. Branicky_, Aug 17 2022