A368020 Palindromes which are a concatenation of three palindromes, each of which has at least 2 digits.

110011, 111111, 112211, 113311, 114411, 115511, 116611, 117711, 118811, 119911, 220022, 221122, 222222, 223322, 224422, 225522, 226622, 227722, 228822, 229922, 330033, 331133, 332233, 333333, 334433, 335533, 336633, 337733, 338833, 339933, 440044, 441144, 442244

Offset

1,1

Comments

Equivalently, these are palindromes which have a palindromic prefix of length at least 2 and no more than 1 less than half the total length. For example, 7 digit terms have the form (aa)(bcb)(aa) and 8 digit terms are of the form (aa)(bccb)(aa) or (aba)(cc)(aba).

Links

Table of n, a(n) for n=1..33.

James S. DeArmon, Python code

Example

110011 is a term since it is a palindrome, and consists of 3 palindromes: (11)(00)(11).
9999999 is a term and its constituent 3 palindromes can be listed in three ways: (99)(999)(99), (999)(99)(99), and (99)(99)(999).

Prog

(Python) # see Link
(Python)
from itertools import count, islice, product
def pals(d=2): # generator of palindromes with d >=2 digits as strings
    yield from (f+(s:="".join(r))+m+s[::-1]+f for f in "123456789" for r in product("0123456789", repeat=d//2-1) for m in [[""], "0123456789"][d%2])
def agen(): # generator of terms
    yield from (int("".join(p)) for d in count(6) for p in pals(d) if any((s:=p[:i])==s[::-1] for i in range(2, d//2)))
print(list(islice(agen(), 33))) # Michael S. Branicky, Jan 23 2024

Crossrefs

Cf. A002113 (palindromes), A344550.

Sequence in context: A111888 A183770 A023349 * A138722 A122240 A250591

Adjacent sequences: A368017 A368018 A368019 * A368021 A368022 A368023

Keyword

base,nonn

Author

James S. DeArmon, Dec 23 2023

Status

approved