A057890 In base 2, either a palindrome or becomes a palindrome if trailing 0's are omitted.

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, 34, 36, 40, 42, 45, 48, 51, 54, 56, 60, 62, 63, 64, 65, 66, 68, 72, 73, 80, 84, 85, 90, 93, 96, 99, 102, 107, 108, 112, 119, 120, 124, 126, 127, 128, 129, 130, 132, 136, 144, 146

Offset

1,3

Comments

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

Fixed points of A057889, complement of A057891

n such that A000265(n) is in A006995. - Robert Israel, Jun 07 2016

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Aayush Rajasekaran, Jeffrey Shallit, and Tim Smith, Sums of Palindromes: an Approach via Nested-Word Automata, preprint arXiv:1706.10206 [cs.FL], June 30 2017.

Formula

A030101(A030101(n)) = A030101(n). - David W. Wilson, Jun 09 2009, Jun 18 2009

A178225(A000265(a(n))) = 1. - Reinhard Zumkeller, Oct 21 2011

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

Example

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

Maple

dmax:= 10: # to get all terms < 2^dmax
revdigs:= proc(n)
  local L, Ln, i;
  L:= convert(n, base, 2);
  Ln:= nops(L);
  add(L[i]*2^(Ln-i), i=1..Ln);
end proc;
P[0]:= {0}:
P[1]:= {1}:
for d from 2 to dmax do
  if d::even then
    P[d]:= { seq(2^(d/2)*x + revdigs(x), x=2^(d/2-1)..2^(d/2)-1)}
  else
    m:= (d-1)/2;
    B:={seq(2^(m+1)*x + revdigs(x), x=2^(m-1)..2^m-1)};
    P[d]:= B union map(`+`, B, 2^m)
  fi
od:
A:= `union`(seq(seq(map(`*`, P[d], 2^k), k=0..dmax-d), d=0..dmax)):
sort(convert(A, list)); # Robert Israel, Jun 07 2016

Mathematica

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 *)
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 *)
A057890Q = If[# > 0 && EvenQ@#, #0[#/2], # == #~IntegerReverse~2] &; Select[0~Range~146, A057890Q] (* JungHwan Min, Mar 29 2017 *)
Select[Range[0, 200], PalindromeQ[IntegerDigits[#, 2] /. {b__, 0..} -> {b} ]&] (* Jean-François Alcover, Sep 18 2018 *)

Prog

(Haskell)
a057890 n = a057890_list !! (n-1)
a057890_list = 0 : filter ((== 1) . a178225 . a000265) [1..]
-- Reinhard Zumkeller, Oct 21 2011
(Python)
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
(PARI)
bitrev(n) = subst(Pol(Vecrev(binary(n>>valuation(n, 2))), 'x), 'x, 2);
is(n) = my(x = n >> valuation(n, 2)); x == bitrev(x);
concat(0, select(is, vector(147, n, n)))  \\ Gheorghe Coserea, Jun 07 2016
(PARI) is(n)=n==0 || Vecrev(n=binary(n>>valuation(n, 2)))==n \\ Charles R Greathouse IV, Aug 25 2016

Crossrefs

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

Sequence in context: A235028 A062014 A164707 * A161604 A125121 A333762

Adjacent sequences: A057887 A057888 A057889 * A057891 A057892 A057893

Keyword

easy,nonn,base,nice

Author

Marc LeBrun, Sep 25 2000

Status

approved