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Search: id:A006995
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| A006995 |
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Numbers whose binary expansion is palindromic.
(Formerly M2403)
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+0
63
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| 0, 1, 3, 5, 7, 9, 15, 17, 21, 27, 31, 33, 45, 51, 63, 65, 73, 85, 93, 99, 107, 119, 127, 129, 153, 165, 189, 195, 219, 231, 255, 257, 273, 297, 313, 325, 341, 365, 381, 387, 403, 427, 443, 455, 471, 495, 511, 513, 561, 585, 633, 645, 693, 717, 765, 771, 819, 843
(list; graph; listen)
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OFFSET
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1,3
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
Index entries for sequences related to binary expansion of n
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FORMULA
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Comments from Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Dec 31 2008 (Start): Written as a decimal, a(10^n) has 2*n digits. For n>1, the decimal expansion of a(10^n) starts with 22..., 23... or 24...
a(1000)=249903, a(10^10)=24502928886295666773.
An inequality valid for n>1: (2/9)*n^2<a(n)<(1/4)*(n+1)^2.
lim sup a(n)/n^2=1/4 for n-->oo.
lim inf a(n)/n^2=2/9 for n-->oo.
Set k:=k(n):=floor(log_2(n)), i:=i(n):=n-2^k+1, j:=j(n):=i-2^(k-1). Then
a(n)=2^(2k-q)+1+2^p*a(m), where p, q and m are determined as follows:
Case 1: If n+1=2^k, then set p=0, q=2, m=0;
Case 2: If 2^k<n+1<2^k+2^(k-1), then set p=k- floor(log_2(i)), q=2, m=2^floor(log_2(i))+i-1;
Case 3: If n+1=2^k+2^(k-1), then set p=0, q=1, m=0;
Case 4: If 2^k+2^(k-1)<n+1<2^(k+1), then set p=k- floor(log_2(j))-1, q=1, m=2^floor(log_2(i))+i-1;
a(2^n)=2^(2n-2)+2^(n-1)+1; a(2^n-2)=2^(2n-2)-1; a(2^n-1)=2^(2n-2)+1; (End)
A030101(n) = n. - David Wilson, Jun 09 2009
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MATHEMATICA
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palQ[n_Integer, base_Integer] := Module[{idn=IntegerDigits[n, base]}, idn==Reverse[idn]]; then to generate any base b sequence: Select[Range[1000], palQ[ #, b]&]
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PROGRAM
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(PARI) for(n=1, 1000, l=length(binary(n)); b=binary(n); if(sum(i=1, l, abs(component(b, i)-component(b, l+1-i)))==0, print1(n, ", ")))
(PARI) for(i=0, 999, if(vecextract(t=binary(i), "-1..1")==t, print1(i", "))) - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Dec 17 2007
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CROSSREFS
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Palindromes in bases 2 through 10: A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955, A002113.
Sequence in context: A029608 A145388 A121820 this_sequence A163410 A064896 A076188
Adjacent sequences: A006992 A006993 A006994 this_sequence A006996 A006997 A006998
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KEYWORD
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base,easy,nonn,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com), Robert G. Wilson v (rgwv(AT)rgwv.com), Mira Bernstein, L. J. Upton
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EXTENSIONS
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A-number in formula corrected by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 18 2009
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