

A038573


a(n) = 2^A000120(n)  1.


33



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
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OFFSET

0,4


COMMENTS

Essentially the same sequence as A001316, which has much more information, and also A159913.  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
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)
From Omar E. Pol, Jan 24 2016: (Start)
Partial sums give A267700.
a(n) is also the number of cells turned ON at nth generation of the cellular automaton of A267700 in a 90degree sector on the square grid.
a(n) is also the number of Ytoothpicks added at nth generation of the structure of A267700 in a 120degree sector on the triangular grid.
(End)


LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..8191 (terms 0..1023 from T. D. Noe)
David Applegate, The movie version
Michael Gilleland, Some SelfSimilar Integer Sequences
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata
Index entries for sequences related to toothpick sequences
Index entries for sequences that are fixed points of mappings


FORMULA

a(2n) = a(n), a(2n+1) = 2*a(n)+1, a(0) = 0. a(n) = A001316(n)1 = 2^A000120(n)  1.  Daniele Parisse (daniele.parisse(AT)m.dasa.de)
a(n) = number of positive integers k < n such that n XOR k = nk (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)).  Reinhard Zumkeller, Nov 21 2009
a(n) = (n mod 2 + 1) * a(floor(n/2)) + n mod 2.  Reinhard Zumkeller, Oct 10 2012
a(n) = Sum_{i=1..n} C(n,i) mod 2.  Wesley Ivan Hurt, Nov 17 2017
G.f.: 1/(1  x) + Product_{k>=0} (1 + 2*x^(2^k)).  Ilya Gutkovskiy, Aug 20 2019


EXAMPLE

9 = 1001 > 0011 > 3, so a(9)=3.
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


MAPLE

seq(2^convert(convert(n, base, 2), `+`)1, n=0..100); # Robert Israel, Jan 24 2016


MATHEMATICA

Array[ 2^Count[ IntegerDigits[ #, 2 ], 1 ]1&, 100 ]
Nest[ Flatten[ # /. a_Integer > {a, 2a + 1}] &, {0}, 7] (* Robert G. Wilson v, Jan 24 2006 *)


PROG

(PARI) a(n)=2^subst(Pol(binary(n)), x, 1)1
(PARI) a(n) = 2^hammingweight(n)1; \\ Michel Marcus, Jan 24 2016
(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


CROSSREFS

Cf. A007313, A115378.
This is Guy Steele's sequence GS(3, 6) (see A135416).
Cf. also A000079, A001316, A027649, A053581, A159913, A267700.
Write n in bary, sort digits into increasing order: this sequence (b=2), A038574 (b=3), A319652 (b=4), A319653 (b=5), A319654 (b=6), A319655 (b=7), A319656 (b=8), A319657 (b=9), A004185 (b=10).
Sequence in context: A005885 A205145 A061892 * A246591 A173465 A151837
Adjacent sequences: A038570 A038571 A038572 * A038574 A038575 A038576


KEYWORD

nonn,easy,nice


AUTHOR

Marc LeBrun


EXTENSIONS

More terms from Erich Friedman
New definition from N. J. A. Sloane, Mar 01 2008


STATUS

approved



