A273245 -id:A273245 - cp's OEIS frontend

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

Marc LeBrun, Sep 25 2000

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

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

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.

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

a057890 n = a057890_list !! (n-1)
a057890_list = 0 : filter ((== 1) . a178225 . a000265) [1..]
-- Reinhard Zumkeller, Oct 21 2011

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

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 *)

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

is(n)=n==0 || Vecrev(n=binary(n>>valuation(n,2)))==n \\ Charles R Greathouse IV, Aug 25 2016

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

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

A272670 Numbers whose binary expansion is not palindromic but which when reversed and leading zeros omitted, does form a palindrome.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 24, 28, 30, 32, 34, 36, 40, 42, 48, 54, 56, 60, 62, 64, 66, 68, 72, 80, 84, 90, 96, 102, 108, 112, 120, 124, 126, 128, 130, 132, 136, 144, 146, 160, 168, 170, 180, 186, 192, 198, 204, 214, 216, 224, 238, 240, 248, 252, 254

Offset: 1

N. J. A. Sloane, May 19 2016

Decimal interpretation of A273245. Twice A057890.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A006995, A057890, A273245, A273329, subsequence of A154809.

isPal := proc(L::list)
    local i;
    for i from 1 to nops(L)/2 do
        if op(i,L) <> op(-i,L) then
            return false;
        end if;
    end do:
    true ;
end proc:
isA272670 := proc(n)
    local bdgs ;
    bdgs := convert(n,base,2) ;
    if isPal(bdgs) then
        return false;
    else
        A000265(n) ;
        bdgs := convert(%,base,2) ;
        isPal(bdgs) ;
    end if;
end proc:
for n from 1 to 500 do
    if isA272670(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, May 20 2016

A272670_list = [n for n in range(1,10**4) if bin(n)[2:] != bin(n)[:1:-1] and bin(n)[2:].rstrip('0') == bin(n)[:1:-1].lstrip('0')] # Chai Wah Wu, May 21 2016

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

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A272670 Numbers whose binary expansion is not palindromic but which when reversed and leading zeros omitted, does form a palindrome.

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A273239 Non-palindromic numbers whose reversal is a palindrome.

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