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A035327
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Write n in binary, interchange 0's and 1's.
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16
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1, 0, 1, 0, 3, 2, 1, 0, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Also bitwise XOR of n with the nearest Mersenne number (A000225) larger than or equal to n, for n > 0. (For n = 0, a(0) = -1 as opposed to 1). The advantage of using BitXor instead of BaseForm in the Mathematica program is that the result has a Head of Integer, not BaseForm. - Alonso Delarte (alonso.delarte(AT)gmail.com), Jan 14 2006
For n>0: largest m<=n such that no carry occurs when adding m to n in binary arithmetic: A003817(n+1) = a(n) + n = a(n) XOR n. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 14 2009]
a(0) could be considered to be 0 (it was set so from 2004 to 2008) if the binary representation of zero was chosen to be the empty string. - Jason Kimberley, Sep 19 2011
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REFERENCES
| J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.
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LINKS
| R. Zumkeller, Table of n, a(n) for n = 0..10000 [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 14 2009]
J.-P. Allouche and J. Shallit, The Ring of k-regular Sequences, II
R. Stephan, Divide-and-conquer generating functions. I. Elementary sequences
R. Stephan, Some divide-and-conquer sequences ...
R. Stephan, Table of generating functions
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FORMULA
| a(n) = 2^k - n - 1, where 2^(k-1) < n < 2^k.
a(n+1) = (a(n)+n) mod (n+1); a(0) = 1. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jul 22 2002
G.f.: 1 + 1/(1-x)*sum(k>=0, 2^k*x^2^(k+1)/(1+x^2^k)). - Ralf Stephan, May 06 2003
a(0) = 0, a(2n+1) = 2*a(n), a(2n) = 2*a(n) + 1 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 29 2004
a(n) = number of positive integers k < n such that n XOR k > n. a(n) = n - A006257(n). - Paul D. Hanna (pauldhanna(AT)juno.com), Jan 21 2006
a(n)=2^{1+floor(log[2](n))}-n-1 for n>=1; a(0)=1. [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 19 2008]
a(n) = if n<2 then 1 - n else 2*a(floor(n/2)) + 1 - n mod 2. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 20 2010]
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EXAMPLE
| 8 = 1000 -> 0111 = 111 = 7
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MAPLE
| 1, seq(2^(1+floor(log[2](n)))-n-1, n=1..81); [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 19 2008]
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MATHEMATICA
| Table[BaseForm[FromDigits[(IntegerDigits[i, 2]/.{0->1, 1->0}), 2], 10], {i, 0, 90}]
Table[BitXor[n, 2^IntegerPart[Log[2, n] + 1] - 1], {n, 100}] (* Alonso Delarte (alonso.delarte(AT)gmail.com), Jan 14 2006 *)
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PROG
| (PARI) a(n)=sum(k=1, n, if(bitxor(n, k)>n, 1, 0)) (Hanna)
(MAGMA) A035327:=func<n|n eq 0 select 1 else SequenceToInteger(([1-b:b in IntegerToSequence(n, 2)]), 2)>; // Jason Kimberley, Sep 19 2011
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CROSSREFS
| a(n) = A003817(n) - n, for n>0. Cf. A087734.
Cf. A000225, A006257 (Josephus problem).
Cf. A167831, A167877. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 14 2009]
Cf. A007088, A061601, A171960. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 20 2010]
Sequence in context: A194753 A098825 A111460 * A004444 A204533 A085771
Adjacent sequences: A035324 A035325 A035326 * A035328 A035329 A035330
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KEYWORD
| nonn,easy,base
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Vit Planocka (planocka(AT)mistral.cz), Feb 01 2003
a(0) corrected by Paolo P. Lava (paoloplava(AT)gmail.com), Oct 22 2007
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