This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
The triangle begins: 2; 1, 4; 6; 2, 8; 10; 3, 12; 14; 4, 16; 18; 5, 20; 12, 21, 22; 6, 11, 14, 22, 24; 16, 23, 26; 7, 12, 18, 24, 28; 25, 30, 110; 8, 13, 26, 32, 112; 27, 34, 114; 9, 14, 28, 36, 116; 29, 38, 118; 10, 40; 11, 22, 41, 42; 11, 12, 21, 24, 42, 44; ...
See Sigrist link.
def cdb(n):
s, out = str(n), set()
for l in range(1, len(s)+1):
for i in range(len(s)+1-l):
if s[i] == '0': continue
t = int(s[i:i+l])
out.add(int(s[:i] + str(2*t) + s[i+l:]))
if t&1 == 0: out.add(int(s[:i] + str(t//2) + s[i+l:]))
return sorted(out)
print([c for n in range(1, 25) for c in cdb(n)]) # Michael S. Branicky, Jul 24 2022
Examples of optimal ways to reach 1,2,3,...: 1 1, 2 1, 2, 4, 8, 16, 112, 56, 28, 14, 12, 6, 3 1, 2, 4 5 cannot be reached, ends in 0 or 5 1, 2, 4, 8, 16, 112, 56, 28, 14, 12, 6 1, 2, 4, 8, 16, 112, 56, 28, 14, 7 1, 2, 4, 8, 1, 2, 4, 8, 16, 112, 56, 28, 18, 9. 10 cannot be reached, ends in 0 or 5 1, 2, 4, 8, 16, 112, 56, 28, 24, 22, 11 1, 2, 4, 8, 16, 112, 56, 28, 14, 12 1, 2, 4, 8, 16, 13 1, 2, 4, 8, 16, 112, 56, 28, 14 15 cannot be reached, ends in 0 or 5 1, 2, 4, 8, 16 1, 2, 4, 8, 16, 32, 34, 17 1, 2, 4, 8, 16, 112, 56, 28, 18 1, 2, 4, 8, 16, 32, 34, 38, 19 20 cannot be reached, ends in 0 or 5 ...
After applying Choix de Bruxelles (version 2) 4 times to 1, we have the numbers {1,2,4,8,16}. Applying it a fifth time we get the additional numbers {13,26,32,112}, so a(5) = 112.
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.
# See Conrad link.
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
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.
# See Conrad link.
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
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