%I A038573
%S A038573 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,
%T A038573 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,
%U A038573 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
%N A038573 2^A000120(n)-1.
%C A038573 Essentially the same sequence as A001316, which has much more information. 
               - N. J. A. Sloane, Jun 05 2009
%C A038573 Smallest number with same number of 1's in its binary expansion as n.
%C A038573 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. ...........
%C A038573 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 04 2009: 
               (Start)
%C A038573 As an infinite string, 2^n terms per row starting with "1":
%C A038573 (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;...)
%C A038573 Row sums of that triangle = A027649: (1, 4, 14, 46, 454,...); where the
%C A038573 next row sum = current term of A027649 + next term in finite difference
%C A038573 row of A027649, i.e. (1, 3, 10, 32, 100, 308,...) = A053581. (End)
%H A038573 T. D. Noe, <a href="b038573.txt">Table of n, a(n) for n=0..1023</a>
%H A038573 Michael Gilleland, <a href="selfsimilar.html">Some Self-Similar Integer 
               Sequences</a>
%F A038573 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)).
%F A038573 a(n) = number of positive integers k < n such that n XOR k = n-k (cf. 
               A115378). - Paul D. Hanna (pauldhanna(AT)juno.com), Jan 21 2006
%F A038573 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 (reinhard.zumkeller(AT)gmail.com), 
               Nov 21 2009]
%e A038573 9 = 1001 -> 0011 -> 3, so a(9)=3.
%e A038573 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 04 2009: 
               (Start)
%e A038573 Triangle by rows:
%e A038573 . 1;
%e A038573 . 1, 3;
%e A038573 . 1, 3, 3, 7;
%e A038573 . 1, 3, 3, 7, 3, 7, 7, 15;
%e A038573 . 1, 3, 3, 7, 3, 7, 7, 15, 3, 7, 7, 15, 7, 15, 15, 31;
%e A038573 . ...
%e A038573 Row sums: (1, 4, 14, 46,...) = A026749 = last row terms + new set of 
               terms
%e A038573 such that row 3 = (1, 3, 3, 7,) + (3, 7, 7, 15) = 14 + 32 = A027649(3) 
               + A053581(3). (End)
%e A038573 The rows of this triangle converge to A159913. - N. J. A. Sloane, Jun 
               05 2009
%t A038573 Array[ 2^Count[ IntegerDigits[ #, 2 ], 1 ]-1&, 100 ]
%t A038573 Nest[ Flatten[ # /. a_Integer -> {a, 2a + 1}] &, {0}, 7] (from Robert 
               G. Wilson v (rgwv(at)rgwv.com), Jan 24 2006)
%o A038573 (PARI) a(n)=2^subst(Pol(binary(n)),x,1)-1
%Y A038573 Cf. A007313, A115378.
%Y A038573 This is Guy Steele's sequence GS(3, 6) (see A135416).
%Y A038573 A027649, A053581 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 04 
               2009]
%Y A038573 Cf. A000079. [From Omar E. Pol (info(AT)polprimos.com), Jun 07 2009]
%Y A038573 Sequence in context: A087891 A005885 A061892 this_sequence A151837 A163381 
               A160123
%Y A038573 Adjacent sequences: A038570 A038571 A038572 this_sequence A038574 A038575 
               A038576
%K A038573 nonn,easy,nice,new
%O A038573 0,4
%A A038573 Marc LeBrun (mlb(AT)well.com)
%E A038573 More terms from Erich Friedman (erich.friedman(AT)stetson.edu).
%E A038573 New definition from N. J. A. Sloane (njas(AT)research.att.com), Mar 01 
               2008