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A006995
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Numbers whose binary expansion is palindromic.
(Formerly M2403)
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78
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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; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| A178225(a(n)) = 1; union of A048700 and A048701. [Reinhard Zumkeller, Oct 21 2011]
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REFERENCES
| N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| 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
| Comments from Hieronymus Fischer, Dec 31 2008 and Jan 10 2012 (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(10^4) = 24183069,
a(10^5) = 2258634081,
a(10^6) = 249410097687,
a(10^7) = 24350854001805,
a(10^8) = 2229543293296319,
a(10^9) = 248640535848971067,
a(10^10)= 24502928886295666773.
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.
a(2^n-1)=2^(2n-2)-1;
a(2^n)=2^(2n-2)+1;
a(2^n+1)=2^(2n-2)+2^(n-1)+1;
a(2^n+2^(n-1))=2^(2n-1)+1;
Recursion formula for n>2: a(n)=2^(2k-q)+1+2^p*a(m),
where k:=floor(log_2(n-1)), and p, q and m are determined as follows:
Case 1: If n=2^(k+1), then set p=0, q=0, m=1;
Case 2: If 2^k<n<2^k+2^(k-1), then set i:=n-2^k, p=k-floor(log_2(i))-1, q=2, m=2^floor(log_2(i))+i;
Case 3: If n=2^k+2^(k-1), then set p=0, q=1, m=1;
Case 4: If 2^k+2^(k-1)<n<2^(k+1), then set j:=n-2^k-2^(k-1), p=k-floor(log_2(j))-1, q=1, m=2*2^floor(log_2(j))+j;
(End)
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MATHEMATICA
| palQ[n_Integer, base_Integer] := Module[{idn=IntegerDigits[n, base]}, idn==Reverse[idn]]; Select[Range[1000], palQ[ #, 2]&]
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PROG
| (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, Dec 17 2007
(MAGMA) [n: n in [0..850] | Intseq(n, 2) eq Reverse(Intseq(n, 2))]; // Bruno Berselli, Aug 29 2011
(Haskell)
a006995 n = a006995_list !! (n-1)
a006995_list = 0 : filter ((== 1) . a178225) a005408_list
-- Reinhard Zumkeller, Oct 21 2011
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CROSSREFS
| Palindromes in bases 2 through 10: A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955, A002113.
See A057148 for the binary representations.
Cf. A178225, A005408.
Sequence in context: A145388 A121820 A180204 * A163410 A064896 A076188
Adjacent sequences: A006992 A006993 A006994 * A006996 A006997 A006998
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KEYWORD
| nonn,base,easy,nice
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AUTHOR
| 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
| A-number in formula corrected by R. J. Mathar, Jun 18 2009
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