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A369127 S is a "boomerang sequence". Replace each digit d of S by a base-10 palindrome not yet used that contains d: the sequence S remains identical to itself if we follow each palindrome with a comma.

%I A369127 #42 Mar 12 2024 23:43:59
%S A369127 0,1,2,3,4,5,6,7,8,9,11,101,111,202,121,131,141,151,22,303,212,161,
%T A369127 222,171,181,33,191,313,44,414,515,55,616,232,242,323,404,333,252,717,
%U A369127 262,818,66,919,272,282,292,1001,77,1111,1221,88,1331,343,353,1441,99
%N A369127 S is a "boomerang sequence". Replace each digit d of S by a base-10 palindrome not yet used that contains d: the sequence S remains identical to itself if we follow each palindrome with a comma.
%H A369127 Michael S. Branicky, <a href="/A369127/b369127.txt">Table of n, a(n) for n = 1..10000</a>
%e A369127 As a(1) to a(10) are single-digit palindromes, the replacement leaves the terms a(1) to a(10) as they were.
%e A369127 a(11) = 11 and we must replace the first digit d = 1 of 11 by the smallest base-10 palindrome not yet used; this is 11 (as the palindrome 1 has been used before);
%e A369127 a(11) = 11 and we must replace now the second digit d = 1 of 11 by the smallest base-10 palindrome not yet used; this is 101 (as 1 and 11 have been used before);
%e A369127 a(12) = 101 and we must replace the first digit d = 1 of 101 by the smallest base-10 palindrome not yet used; this is 111;
%e A369127 a(12) = 101 and we must now replace the digit d = 0 of 101 by the smallest base-10 palindrome not yet used; this is 202 (as 0 and 101 have been used before);
%e A369127 a(12) = 101 and we must now replace the last digit d = 1 of 101 by the smallest base-10 palindrome not yet used; this is 121; etc.
%o A369127 (Python)
%o A369127 from collections import deque
%o A369127 from itertools import count, islice
%o A369127 def pals(start=1): # generator of palindromes >= palindrome start
%o A369127     s = str(start)
%o A369127     q, r = divmod(len(s)+1, 2)
%o A369127     for d in count(q):
%o A369127         olst = [1, 0][int(d==q and r==1):]
%o A369127         for offset in olst:
%o A369127             lb = max(1, 10**(d-1)) if d>q or offset!=olst[0] else int(s[:q])
%o A369127             for i in range(lb, 10**d):
%o A369127                 left = str(i)
%o A369127                 yield int(left+left[::-1][offset:])
%o A369127 def agen(): # generator of terms
%o A369127     S = deque([101])
%o A369127     head = list(range(10)) + [11]
%o A369127     yield from head
%o A369127     used = set(head) | {101}
%o A369127     pstart = {d:0 for d in "0123456789"}
%o A369127     while True:
%o A369127         an = S.popleft()
%o A369127         yield an
%o A369127         for d in str(an):
%o A369127             p = next(p for p in pals(start=pstart[d]) if p not in used and d in str(p))
%o A369127             pstart[d] = p
%o A369127             S.append(p)
%o A369127             used.add(p)
%o A369127 print(list(islice(agen(), 57))) # _Michael S. Branicky_, Mar 03 2024
%Y A369127 Cf. A002113, A369603, A369604, A369823, A369824, A369798.
%K A369127 base,nonn
%O A369127 1,3
%A A369127 _Eric Angelini_, Mar 02 2024
%E A369127 More terms from _Michael S. Branicky_, Mar 03 2024