A014190 Palindromes in base 3 (written in base 10).

0, 1, 2, 4, 8, 10, 13, 16, 20, 23, 26, 28, 40, 52, 56, 68, 80, 82, 91, 100, 112, 121, 130, 142, 151, 160, 164, 173, 182, 194, 203, 212, 224, 233, 242, 244, 280, 316, 328, 364, 400, 412, 448, 484, 488, 524, 560, 572, 608, 644, 656, 692, 728, 730, 757

Offset

1,3

Comments

Rajasekaran, Shallit, & Smith prove that this sequence is an additive basis of order (exactly) 3. - Charles R Greathouse IV, May 03 2020

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

Patrick De Geest, Palindromic numbers beyond base 10.

Phakhinkon Phunphayap and Prapanpong Pongsriiam, Estimates for the Reciprocal Sum of b-adic Palindromes, 2019.

Aayush Rajasekaran, Jeffrey Shallit, and Tim Smith, Sums of palindromes: an approach via automata, arXiv:1706.10206 [cs.FL], 2017.

Eric Weisstein's World of Mathematics, Palindromic Number.

Eric Weisstein's World of Mathematics, Ternary.

Index entries for sequences that are an additive basis, order 3.

Formula

Sum_{n>=2} 1/a(n) = 2.61676111... (Phunphayap and Pongsriiam, 2019). - Amiram Eldar, Oct 17 2020

Maple

isA014190 := proc(n)
    local L;
    L := convert(n, base, 3) ;
    ListTools[Reverse](L) = L ;
end proc:
for n from 0 to 500 do
    if isA014190(n) then
        printf("%d, ", n) ;
    end if;
end do: # R. J. Mathar, Jul 07 2015

Mathematica

f[n_, b_] := Module[{i=IntegerDigits[n, b]}, i==Reverse[i]]; lst={}; Do[If[f[n, 3], AppendTo[lst, n]], {n, 1000}]; lst (* Vladimir Joseph Stephan Orlovsky, Jul 08 2009 *)

Prog

(Magma) [n: n in [0..800] | Intseq(n, 3) eq Reverse(Intseq(n, 3))]; // Vincenzo Librandi, Sep 09 2015
(Sage)
[n for n in (0..757) if Word(n.digits(3)).is_palindrome()] # Peter Luschny, Sep 13 2018
(PARI) ispal(n, b=3)=my(d=digits(n, b)); d==Vecrev(d) \\ Charles R Greathouse IV, May 03 2020
(Python)
from gmpy2 import digits
def A014190(n):
    if n == 1: return 0
    y = 3*(x:=3**(len(digits(n>>1, 3))-1))
    return int((c:=n-x)*x+int(digits(c, 3)[-2::-1]or'0', 3) if n<x+y else (c:=n-y)*y+int(digits(c, 3)[-1::-1]or'0', 3)) # Chai Wah Wu, Jun 13 2024

Crossrefs

Cf. A007089, A118594, A134027, A330312 (first differences).

Palindromes in bases 2 through 10: A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955, A002113.

Sequence in context: A100278 A075333 A297250 * A141400 A190744 A190751

Adjacent sequences: A014187 A014188 A014189 * A014191 A014192 A014193

Keyword

nonn,base,easy,changed

Author

N. J. A. Sloane

Status

approved