A356715 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 ternary (base 3).
1, 2, 3, 6, 11, 26, 68, 177, 492, 1403, 4113, 12149, 36225, 108268, 324529, 973163, 2920533, 8764041, 26303715, 78935398, 236878491, 710783343
Offset: 0
Examples
For n = 2, a(2) = 3 since the numbers obtained are (in base 3): 1, 2, 11. For n = 4, they expand to a(5) = 11 numbers (in base 3): 1, 2, 11, 12, 21, 22, 101, 111, 112, 121, 211.
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.
Programs
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Python
# See Conrad link.
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Python
from itertools import islice from sympy.ntheory import digits def fd(d, b): return sum(b**i*di for i, di in enumerate(d[::-1])) def cdb2(n, base=3): d, out = digits(n, base)[1:], {n} for l in range(1, len(d)+1): for i in range(len(d)+1-l): if d[i] == 0: continue t = fd(d[i:i+l], base) out.add(fd(d[:i] + digits(2*t, base)[1:] + d[i+l:], base)) if t&1 == 0: out.add(fd(d[:i] + digits(t//2, base)[1:] + d[i+l:], base)) return out def agen(): reach, expand = {1}, [1] while True: yield len(reach) #; print(reach); print([digits(t, 3)[1:] for t in sorted(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(), 10))) # Michael S. Branicky, Aug 24 2022
Extensions
a(15)-a(19) from Michael S. Branicky, Aug 24 2022
a(20)-a(21) from Michael S. Branicky, Aug 30 2022