A369798 S is a "boomerang sequence": multiply each digit d of S by the number to which d belongs: the sequence S remains identical to itself if we follow each multiplication with a comma.

0, 1, 12, 24, 48, 96, 192, 384, 864, 576, 192, 1728, 384, 1152, 3072, 1536, 6912, 5184, 3456, 2880, 4032, 3456, 192, 1728, 384, 1728, 12096, 3456, 13824, 1152, 3072, 1536, 1152, 1152, 5760, 2304, 9216, 0, 21504, 6144, 1536, 7680, 4608, 9216, 41472, 62208, 6912, 13824, 25920, 5184, 41472, 20736, 10368, 13824

Offset: 1

Author

Eric Angelini and Jean-Marc Falcoz, Feb 01 2024

Keywords

• base
• nonn

Comments

S is the lexicographycally earliest nontrivial sequence of nonnegative integers with this property (if we try for a(3) the integers 1, 10 or 11, we respectively get these trivial sequences):

S = 1, 1, 1, 1, 1, 1, 1, ...

S = 1, 10, 0, 0, 0, 0, 0, ...

S = 1, 11, 1, 1, 1, 1, 1, ...

Examples

			a(1) = 0, which multiplied by 0 gives 0
a(2) = 1, which multiplied by 1 gives 1
a(3) = 12
     1st digit is 1, which multiplied by 12 gives 12
     2nd digit is 2, which multiplied by 12 gives 24
a(4) = 24
     1st digit is 2, which multiplied by 24 gives 48
     2nd digit is 4, which multiplied by 24 gives 96
a(5) = 48
     1st digit is 4, which multiplied by 48 gives 192
     2nd digit is 8, which multiplied by 48 gives 384
a(6) = 96
     1st digit is 9, which multiplied by 96 gives 864
     2nd digit is 6, which multiplied by 96 gives 576
Etc. We see that the above last column reproduces S.
		

Links

• Eric Angelini and Jean-Marc Falcoz, Boomerang sequences, Personal blog, Feb 1st 2024.

Crossrefs

Cf. A369603, A369604, A369823, A369824.

Programs

Mathematica

Join[{0,1},Nest[Flatten[IntegerDigits@#*#]&,{12},5]] (* Giorgos Kalogeropoulos, Feb 01 2024 *)

Python

from itertools import islice
from collections import deque
def agen(): # generator of terms
    S = deque([24])
    yield from [0, 1, 12]
    while True:
        an = S.popleft()
        yield an
        S.extend(an*d for d in map(int, str(an)))
print(list(islice(agen(), 54))) # Michael S. Branicky, Feb 01 2024