A057890 - OEIS

%I #61 Dec 07 2019 12:18:22

%S 0,1,2,3,4,5,6,7,8,9,10,12,14,15,16,17,18,20,21,24,27,28,30,31,32,33,

%T 34,36,40,42,45,48,51,54,56,60,62,63,64,65,66,68,72,73,80,84,85,90,93,

%U 96,99,102,107,108,112,119,120,124,126,127,128,129,130,132,136,144,146

%N In base 2, either a palindrome or becomes a palindrome if trailing 0's are omitted.

%C Symmetric bit strings (bit-reverse palindromes), including as many leading as trailing zeros.

%C Fixed points of A057889, complement of A057891

%C n such that A000265(n) is in A006995. - _Robert Israel_, Jun 07 2016

%H Reinhard Zumkeller, <a href="/A057890/b057890.txt">Table of n, a(n) for n = 1..10000</a>

%H Aayush Rajasekaran, Jeffrey Shallit, and Tim Smith, <a href="https://arxiv.org/abs/1706.10206">Sums of Palindromes: an Approach via Nested-Word Automata</a>, preprint arXiv:1706.10206 [cs.FL], June 30 2017.

%F A030101(A030101(n)) = A030101(n). - _David W. Wilson_, Jun 09 2009, Jun 18 2009

%F A178225(A000265(a(n))) = 1. - _Reinhard Zumkeller_, Oct 21 2011

%F a(7*2^n-4*n-4) = 4^n + 1, a(10*2^n-4*n-6) = 2*4^n + 1. - _Gheorghe Coserea_, Apr 05 2017

%e 10 is included, since 01010 is a palindrome, but 11 is not because 1011 is not.

%p dmax:= 10: # to get all terms < 2^dmax

%p revdigs:= proc(n)

%p   local L, Ln, i;

%p   L:= convert(n, base, 2);

%p   Ln:= nops(L);

%p   add(L[i]*2^(Ln-i), i=1..Ln);

%p end proc;

%p P[0]:= {0}:

%p P[1]:= {1}:

%p for d from 2 to dmax do

%p   if d::even then

%p     P[d]:= { seq(2^(d/2)*x + revdigs(x), x=2^(d/2-1)..2^(d/2)-1)}

%p   else

%p     m:= (d-1)/2;

%p     B:={seq(2^(m+1)*x + revdigs(x), x=2^(m-1)..2^m-1)};

%p     P[d]:= B union map(`+`, B, 2^m)

%p   fi

%p od:

%p A:= `union`(seq(seq(map(`*`,P[d],2^k),k=0..dmax-d),d=0..dmax)):

%p sort(convert(A,list)); # _Robert Israel_, Jun 07 2016

%t PaleQ[n_Integer, base_Integer] := Module[{idn, trim = n/base^IntegerExponent[n, base]}, idn = IntegerDigits[trim, base]; idn == Reverse[idn]]; Select[Range[0, 150], PaleQ[#, 2] &] (* _Lei Zhou_, Dec 13 2013 *)

%t pal2Q[n_]:=Module[{id=Drop[IntegerDigits[n,2],-IntegerExponent[n,2]]},id==Reverse[id]]; Join[{0},Select[Range[200],pal2Q]] (* _Harvey P. Dale_, Feb 26 2015 *)

%t A057890Q = If[# > 0 && EvenQ@#, #0[#/2], # == #~IntegerReverse~2] &; Select[0~Range~146, A057890Q] (* _JungHwan Min_, Mar 29 2017 *)

%t Select[Range[0, 200], PalindromeQ[IntegerDigits[#, 2] /. {b__, 0..} -> {b} ]&] (* _Jean-François Alcover_, Sep 18 2018 *)

%o (Haskell)

%o a057890 n = a057890_list !! (n-1)

%o a057890_list = 0 : filter ((== 1) . a178225 . a000265) [1..]

%o -- _Reinhard Zumkeller_, Oct 21 2011

%o (Python)

%o A057890 = [n for n in range(10**6) if bin(n)[2:].rstrip('0') == bin(n)[2:].rstrip('0')[::-1]] # _Chai Wah Wu_, Aug 12 2014

%o (PARI)

%o bitrev(n) = subst(Pol(Vecrev(binary(n>>valuation(n,2))), 'x), 'x, 2);

%o is(n) = my(x = n >> valuation(n,2)); x == bitrev(x);

%o concat(0, select(is,vector(147,n,n)))  \\ _Gheorghe Coserea_, Jun 07 2016

%o (PARI) is(n)=n==0 || Vecrev(n=binary(n>>valuation(n,2)))==n \\ _Charles R Greathouse IV_, Aug 25 2016

%Y Cf. A030101, A000265, A006519, A006995, A057889, A057891, A061917, A273245, A273329, A272670.

%K easy,nonn,base,nice

%O 1,3

%A _Marc LeBrun_, Sep 25 2000