A369798 - OEIS

%I #30 Feb 05 2024 10:49:41

%S 0,1,12,24,48,96,192,384,864,576,192,1728,384,1152,3072,1536,6912,

%T 5184,3456,2880,4032,3456,192,1728,384,1728,12096,3456,13824,1152,

%U 3072,1536,1152,1152,5760,2304,9216,0,21504,6144,1536,7680,4608,9216,41472,62208,6912,13824,25920,5184,41472,20736,10368,13824

%N 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.

%C 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):

%C S = 1, 1, 1, 1, 1, 1, 1, ...

%C S = 1, 10, 0, 0, 0, 0, 0, ...

%C S = 1, 11, 1, 1, 1, 1, 1, ...

%H Eric Angelini and Jean-Marc Falcoz, <a href="https://cinquantesignes.blogspot.com/2024/02/boomerang-sequences.html">Boomerang sequences</a>, Personal blog, Feb 1st 2024.

%e a(1) = 0, which multiplied by 0 gives 0

%e a(2) = 1, which multiplied by 1 gives 1

%e a(3) = 12

%e      1st digit is 1, which multiplied by 12 gives 12

%e      2nd digit is 2, which multiplied by 12 gives 24

%e a(4) = 24

%e      1st digit is 2, which multiplied by 24 gives 48

%e      2nd digit is 4, which multiplied by 24 gives 96

%e a(5) = 48

%e      1st digit is 4, which multiplied by 48 gives 192

%e      2nd digit is 8, which multiplied by 48 gives 384

%e a(6) = 96

%e      1st digit is 9, which multiplied by 96 gives 864

%e      2nd digit is 6, which multiplied by 96 gives 576

%e Etc. We see that the above last column reproduces S.

%t Join[{0,1},Nest[Flatten[IntegerDigits@#*#]&,{12},5]] (* _Giorgos Kalogeropoulos_, Feb 01 2024 *)

%o (Python)

%o from itertools import islice

%o from collections import deque

%o def agen(): # generator of terms

%o     S = deque([24])

%o     yield from [0, 1, 12]

%o     while True:

%o         an = S.popleft()

%o         yield an

%o         S.extend(an*d for d in map(int, str(an)))

%o print(list(islice(agen(), 54))) # _Michael S. Branicky_, Feb 01 2024

%Y Cf. A369603, A369604, A369823, A369824.

%K base,nonn

%O 1,3

%A _Eric Angelini_ and _Jean-Marc Falcoz_, Feb 01 2024