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%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