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A057890 In base 2, either a palindrome or becomes a palindrome if trailing 0's are omitted. 28
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 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A235028 A062014 A164707 * A161604 A125121 A333762
KEYWORD
easy,nonn,base,nice
AUTHOR
Marc LeBrun, Sep 25 2000
STATUS
approved

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Last modified March 19 09:40 EDT 2024. Contains 370981 sequences. (Running on oeis4.)