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A029956 Numbers that are palindromic in base 11.

%I A029956 #39 Jun 14 2024 11:55:30
%S A029956 0,1,2,3,4,5,6,7,8,9,10,12,24,36,48,60,72,84,96,108,120,122,133,144,
%T A029956 155,166,177,188,199,210,221,232,244,255,266,277,288,299,310,321,332,
%U A029956 343,354,366,377,388,399,410,421,432,443,454,465,476,488,499
%N A029956 Numbers that are palindromic in base 11.
%C A029956 Cilleruelo, Luca, & Baxter prove that this sequence is an additive basis of order (exactly) 3. - _Charles R Greathouse IV_, May 04 2020
%H A029956 John Cerkan, <a href="/A029956/b029956.txt">Table of n, a(n) for n = 1..10000</a>
%H A029956 Javier Cilleruelo, Florian Luca and Lewis Baxter, <a href="https://doi.org/10.1090/mcom/3221">Every positive integer is a sum of three palindromes</a>, Mathematics of Computation, Vol. 87, No. 314 (2018), pp. 3023-3055, <a href="http://arxiv.org/abs/1602.06208">arXiv preprint</a>, arXiv:1602.06208 [math.NT], 2017.
%H A029956 Patrick De Geest, <a href="http://www.worldofnumbers.com/nobase10.htm">Palindromic numbers beyond base 10</a>.
%H A029956 Phakhinkon Phunphayap and Prapanpong Pongsriiam, <a href="https://doi.org/10.13140/RG.2.2.23202.79047">Estimates for the Reciprocal Sum of b-adic Palindromes</a>, 2019.
%H A029956 <a href="/index/Ab#basis_03">Index entries for sequences that are an additive basis</a>, order 3.
%F A029956 Sum_{n>=2} 1/a(n) = 3.4369816... (Phunphayap and Pongsriiam, 2019). - _Amiram Eldar_, Oct 17 2020
%t A029956 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 *)
%t A029956 pal11Q[n_]:=Module[{idn11=IntegerDigits[n,11]},idn11==Reverse[idn11]]; Select[Range[0,500],pal11Q] (* _Harvey P. Dale_, May 11 2015 *)
%t A029956 Select[Range[0, 500], PalindromeQ[IntegerDigits[#, 11]] &] (* _Michael De Vlieger_, May 12 2017, Version 10.3 *)
%o A029956 (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
%o A029956 is(n)=ispal(n,11) \\ _Charles R Greathouse IV_, Aug 21 2012
%o A029956 (PARI) ispal(n,b=11)=my(d=digits(n,b)); d==Vecrev(d) \\ _Charles R Greathouse IV_, May 04 2020
%o A029956 (Sage)
%o A029956 [n for n in (0..499) if Word(n.digits(11)).is_palindrome()] # _Peter Luschny_, Sep 13 2018
%o A029956 (Python)
%o A029956 from gmpy2 import digits
%o A029956 from sympy import integer_log
%o A029956 def A029956(n):
%o A029956     if n == 1: return 0
%o A029956     y = 11*(x:=11**integer_log(n>>1,11)[0])
%o A029956     return int((c:=n-x)*x+int(digits(c,11)[-2::-1]or'0',11) if n<x+y else (c:=n-y)*y+int(digits(c,11)[-1::-1]or'0',11)) # _Chai Wah Wu_, Jun 14 2024
%Y A029956 Cf. A002113 (base 10), A029957 (base 12).
%K A029956 nonn,base,easy
%O A029956 1,3
%A A029956 _Patrick De Geest_