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0, 1, 1, 3, 1, 3, 3, 7, 1, 3, 3, 7, 3, 7, 7, 15, 1, 3, 3, 7, 3, 7, 7, 15, 3, 7, 7, 15, 7, 15, 15, 31, 1, 3, 3, 7, 3, 7, 7, 15, 3, 7, 7, 15, 7, 15, 15, 31, 3, 7, 7, 15, 7, 15, 15, 31, 7, 15, 15, 31, 15, 31, 31, 63, 1, 3, 3, 7, 3, 7, 7, 15, 3, 7, 7, 15, 7, 15, 15, 31, 3, 7, 7, 15, 7, 15, 15, 31
(list;
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refs;
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history;
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internal format)
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OFFSET
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0,4
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COMMENTS
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Essentially the same sequence as A001316, which has much more information. - N. J. A. Sloane, Jun 05 2009
Smallest number with same number of 1's in its binary expansion as n.
Fixed point of the morphism 0 -> 01, 1 -> 13, 3 -> 37, ... = k -> k, 2k+1, ... starting from a(0) = 0; 1 -> 01 -> 0113 -> 01131337 -> 011313371337377(15) -> ..., . - Robert G. Wilson v Jan 24 2006. ...........
Contribution from Gary W. Adamson, Jun 04 2009: (Start)
As an infinite string, 2^n terms per row starting with "1":
(1; 1,3; 1,3,3,7; 1,3,3,7,3,7,7,15; 1,3,3,7,3,7,7,15,3,7,7,15,7,15,15,3l;...)
Row sums of that triangle = A027649: (1, 4, 14, 46, 454,...); where the
next row sum = current term of A027649 + next term in finite difference
row of A027649, i.e. (1, 3, 10, 32, 100, 308,...) = A053581. (End)
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..1023
Michael Gilleland, Some Self-Similar Integer Sequences
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FORMULA
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a(2n) = a(n), a(2n+1) = 2*a(n)+1, a(0) = 0. a(n) = A001316(n)-1 = 2^A000120(n)-1 (comment from Daniele Parisse (daniele.parisse(AT)m.dasa.de)).
a(n) = number of positive integers k < n such that n XOR k = n-k (cf. A115378). - Paul D. Hanna, Jan 21 2006
a(n) = f(n, 1) with f(x, y) = if x = 0 then y - 1 else f(floor(x/2), y*(1 + x mod 2)). [From Reinhard Zumkeller, Nov 21 2009]
a(n) = (n mod 2 + 1) * a(floor(n/2)) + n mod 2. - Reinhard Zumkeller, Oct 10 2012
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EXAMPLE
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9 = 1001 -> 0011 -> 3, so a(9)=3.
Contribution from Gary W. Adamson, Jun 04 2009: (Start)
Triangle read by rows:
. 1;
. 1, 3;
. 1, 3, 3, 7;
. 1, 3, 3, 7, 3, 7, 7, 15;
. 1, 3, 3, 7, 3, 7, 7, 15, 3, 7, 7, 15, 7, 15, 15, 31;
. ...
Row sums: (1, 4, 14, 46,...) = A026749 = last row terms + new set of terms
such that row 3 = (1, 3, 3, 7,) + (3, 7, 7, 15) = 14 + 32 = A027649(3) + A053581(3). (End)
The rows of this triangle converge to A159913. - N. J. A. Sloane, Jun 05 2009
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MATHEMATICA
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Array[ 2^Count[ IntegerDigits[ #, 2 ], 1 ]-1&, 100 ]
Nest[ Flatten[ # /. a_Integer -> {a, 2a + 1}] &, {0}, 7] (from Robert G. Wilson v, Jan 24 2006)
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PROG
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(PARI) a(n)=2^subst(Pol(binary(n)), x, 1)-1
(Haskell)
a038573 0 = 0
a038573 n = (m + 1) * (a038573 n') + m where (n', m) = divMod n 2
-- Reinhard Zumkeller, Oct 10 2012, Feb 07 2011
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CROSSREFS
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Cf. A007313, A115378.
This is Guy Steele's sequence GS(3, 6) (see A135416).
A027649, A053581 [From Gary W. Adamson, Jun 04 2009]
Cf. A000079. [From Omar E. Pol, Jun 07 2009]
Sequence in context: A005885 A205145 A061892 * A173465 A151837 A163381
Adjacent sequences: A038570 A038571 A038572 * A038574 A038575 A038576
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KEYWORD
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nonn,easy,nice
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AUTHOR
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Marc LeBrun
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EXTENSIONS
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More terms from Erich Friedman.
New definition from N. J. A. Sloane, Mar 01 2008
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STATUS
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approved
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