A029956 Numbers that are palindromic in base 11.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 122, 133, 144, 155, 166, 177, 188, 199, 210, 221, 232, 244, 255, 266, 277, 288, 299, 310, 321, 332, 343, 354, 366, 377, 388, 399, 410, 421, 432, 443, 454, 465, 476, 488, 499

Offset: 1

Author

Patrick De Geest

Keywords

• nonn
• base
• easy

Comments

Cilleruelo, Luca, & Baxter prove that this sequence is an additive basis of order (exactly) 3. - Charles R Greathouse IV, May 04 2020

Links

• John Cerkan, Table of n, a(n) for n = 1..10000
• Javier Cilleruelo, Florian Luca and Lewis Baxter, Every positive integer is a sum of three palindromes, Mathematics of Computation, Vol. 87, No. 314 (2018), pp. 3023-3055, arXiv preprint, arXiv:1602.06208 [math.NT], 2017.
• Patrick De Geest, Palindromic numbers beyond base 10.
• Phakhinkon Phunphayap and Prapanpong Pongsriiam, Estimates for the Reciprocal Sum of b-adic Palindromes, 2019.
• Index entries for sequences that are an additive basis, order 3.

Crossrefs

Cf. A002113 (base 10), A029957 (base 12).

Programs

Mathematica

f[n_,b_]:=Module[{i=IntegerDigits[n,b]},i==Reverse[i]];lst={};Do[If[f[n,11],AppendTo[lst,n]],{n,7!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 08 2009 *)
pal11Q[n_]:=Module[{idn11=IntegerDigits[n,11]},idn11==Reverse[idn11]]; Select[Range[0,500],pal11Q] (* Harvey P. Dale, May 11 2015 *)
Select[Range[0, 500], PalindromeQ[IntegerDigits[#, 11]] &] (* Michael De Vlieger, May 12 2017, Version 10.3 *)

PARI

ispal(n,b)=my(tmp,d=log(n+.5)\log(b)-1);while(d,tmp=n%b;n\=b;if(n\b^d!=tmp,return(0));n=n%(b^d);d-=2;);d<0||n%(b+1)==0
is(n)=ispal(n,11) \\ Charles R Greathouse IV, Aug 21 2012

PARI

ispal(n,b=11)=my(d=digits(n,b)); d==Vecrev(d) \\ Charles R Greathouse IV, May 04 2020

Python

from gmpy2 import digits
from sympy import integer_log
def A029956(n):
    if n == 1: return 0
    y = 11*(x:=11**integer_log(n>>1,11)[0])
    return int((c:=n-x)*x+int(digits(c,11)[-2::-1]or'0',11) if nChai Wah Wu, Jun 14 2024

Sage

[n for n in (0..499) if Word(n.digits(11)).is_palindrome()] # Peter Luschny, Sep 13 2018

Formula

Sum_{n>=2} 1/a(n) = 3.4369816... (Phunphayap and Pongsriiam, 2019). - Amiram Eldar, Oct 17 2020