A118594 Palindromes in base 3 (written in base 3).

0, 1, 2, 11, 22, 101, 111, 121, 202, 212, 222, 1001, 1111, 1221, 2002, 2112, 2222, 10001, 10101, 10201, 11011, 11111, 11211, 12021, 12121, 12221, 20002, 20102, 20202, 21012, 21112, 21212, 22022, 22122, 22222, 100001, 101101, 102201, 110011, 111111, 112211, 120021

Offset: 1

Author

Martin Renner, May 08 2006

Keywords

• nonn
• base
• easy

Comments

The number of n-digit terms is given by A225367. - M. F. Hasler, May 05 2013 [Moved here on May 08 2013]

Digit-wise application of A000578 (and also superposition of a(n) with its horizontal OR vertical reflection) yields A006072. - M. F. Hasler, May 08 2013

Equivalently, palindromes k (written in base 10) such that 4*k is a palindrome. - Bruno Berselli, Sep 12 2018

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Palindromic Number
• Eric Weisstein's World of Mathematics, Ternary

Crossrefs

Cf. A007089, A014190, A057148, A118595, A118596, A118597, A118598, A118599, A118600, A002113.

Programs

Mathematica

(* get NextPalindrome from A029965 *) Select[NestList[NextPalindrome, 0, 1110], Max@IntegerDigits@# < 3 &] (* Robert G. Wilson v, May 09 2006 *)
Select[FromDigits/@Tuples[{0,1,2},8],IntegerDigits[#]==Reverse[ IntegerDigits[ #]]&] (* Harvey P. Dale, Apr 20 2015 *)

PARI

{for(l=1,5,u=vector((l+1)\2,i,10^(i-1)+(2*i-11&&i==1,2]), print1(v*u",")))} \\ The n-th term could be produced by using (partial sums of) A225367 to skip all shorter terms, and then skipping the adequate number of vectors v until n is reached.  - M. F. Hasler, May 08 2013

Python

from itertools import count, islice, product
def agen(): # generator of terms
    yield from [0, 1, 2]
    for d in count(2):
        for start in "12":
            for rest in product("012", repeat=d//2-1):
                left = start + "".join(rest)
                for mid in [[""], ["0", "1", "2"]][d%2]:
                    yield int(left + mid + left[::-1])
print(list(islice(agen(), 42))) # Michael S. Branicky, Mar 29 2022

Python

from sympy import integer_log
from gmpy2 import digits
def A118594(n):
    if n == 1: return 0
    y = 3*(x:=3**integer_log(n>>1,3)[0])
    return int((s:=digits(n-x,3))+s[-2::-1] if nChai Wah Wu, Jun 14 2024

Sage

[int(n.str(base=3)) for n in (0..757) if Word(n.digits(3)).is_palindrome()] # Peter Luschny, Sep 13 2018

Extensions

More terms from Robert G. Wilson v, May 09 2006

a(40) and beyond from Michael S. Branicky, Mar 29 2022