A356177 Palindromes in A333369.

1, 3, 5, 7, 9, 22, 44, 66, 88, 111, 212, 232, 252, 272, 292, 333, 414, 434, 454, 474, 494, 555, 616, 636, 656, 676, 696, 777, 818, 838, 858, 878, 898, 999, 2002, 2222, 2442, 2662, 2882, 4004, 4224, 4444, 4664, 4884, 6006, 6226, 6446, 6666, 6886, 8008, 8228, 8448, 8668, 8888, 10101

Offset

1,2

Comments

If a term has a decimal digit that is odd, it must have an odd number of decimal digits and all odd digits are the same. - Chai Wah Wu, Jul 29 2022

If a term has an even number of decimal digits, then it must have only even decimal digits. - Bernard Schott, Jul 30 2022

Links

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Example

474 is palindrome and 474 has two 4's and one 7 in its decimal expansion, hence 474 is a term.

Mathematica

simQ[n_] := AllTrue[Tally @ IntegerDigits[n], EvenQ[Plus @@ #] &]; Select[Range[10^4], PalindromeQ[#] && simQ[#] &] (* Amiram Eldar, Jul 28 2022 *)

Prog

(Python)
from itertools import count, islice, product
def simb(n): s = str(n); return all(s.count(d)%2==int(d)%2 for d in set(s))
def pals(): # generator of palindromes
    digits = "0123456789"
    for d in count(1):
        for p in product(digits, repeat=d//2):
            if d > 1 and p[0] == "0": continue
            left = "".join(p); right = left[::-1]
            for mid in [[""], digits][d%2]:
                yield int(left + mid + right)
def agen(): yield from filter(simb, pals())
print(list(islice(agen(), 55))) # Michael S. Branicky, Jul 28 2022
(Python) # faster version based on Comments
from itertools import count, islice, product
def odgen(d): yield from [1, 3, 5, 7, 9] if d == 1 else sorted(int(f+"".join(p)+o+"".join(p[::-1])+f) for o in "13579" for f in o + "2468" for p in product(o+"02468", repeat=d//2-1))
def evgen(d): yield from (int(f+"".join(p)+"".join(p[::-1])+f) for f in "2468" for p in product("02468", repeat=d//2-1))
def A356177gen():
    for d in count(1, step=2): yield from odgen(d); yield from evgen(d+1)
print(list(islice(A356177gen(), 55))) # Michael S. Branicky, Jul 30 2022

Crossrefs

Intersection of A002113 and A333369.

Cf. A355770, A355771, A355772, A100706 (subsequence of repunits).

Sequence in context: A241175 A342729 A099995 * A329364 A100028 A053681

Adjacent sequences: A356174 A356175 A356176 * A356178 A356179 A356180

Keyword

nonn,base

Author

Bernard Schott, Jul 28 2022

Status

approved