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

1,3

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, ...

Links

Table of n, a(n) for n=1..54.

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

Example

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.

Mathematica

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

Prog

(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

Crossrefs

Cf. A369603, A369604, A369823, A369824.

Sequence in context: A367361 A180617 A081808 * A260261 A080495 A090776

Adjacent sequences: A369795 A369796 A369797 * A369799 A369800 A369801

Keyword

base,nonn

Author

Eric Angelini and Jean-Marc Falcoz, Feb 01 2024

Status

approved