A038573 a(n) = 2^A000120(n) - 1.

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

Offset: 0

Author

Marc LeBrun

Keywords

• nonn
• easy
• nice
• changed

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; 1,3,3,7,3,7,7,15,3,7,7,15,7,15,15,31;...)

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 n-th generation of the cellular automaton of A267700 in a 90-degree sector on the square grid.

a(n) is also the number of Y-toothpicks added at n-th generation of the structure of A267700 in a 120-degree sector on the triangular grid. (End)

Row sums of A090971. - Nikolaos Pantelidis, Nov 23 2022

Examples

			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, ...) = A027649 = last row terms + new set of terms such that row 3 = (1, 3, 3, 7,) + (3, 7, 7, 15) = 14 + 32 = A027649(2) + A053581(3). (End)
The rows of this triangle converge to A159913. - _N. J. A. Sloane_, Jun 05 2009
G.f. = x + x^2 + 3*x^3 + x^4 + 3*x^5 + 3*x^6 + 7*x^7 + x^8 + 3*x^9 + 3*x^10 + 7*x^11 + ... - _Michael Somos_, Jul 24 2023
		

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 Self-Similar 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

Crossrefs

Cf. A007313, A115378, A073138.

This is Guy Steele's sequence GS(3, 6) (see A135416).

Cf. also A000079, A001316, A027649, A053581, A159913, A267700.

Write n in b-ary, 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).

Column k=0 of A340666.

Programs

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
(Python 3.10+)
def A038573(n): return (1<Chai Wah Wu, Nov 15 2022

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 *)

PARI

{a(n) = 2^subst(Pol(binary(n)), x, 1) - 1};

PARI

a(n) = 2^hammingweight(n)-1; \\ Michel Marcus, Jan 24 2016

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

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)). - 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

G.f. A(x) = x + x^2*A(x) + (1 + 2*x)*(1 - x^2)*A(x^2). - Michael Somos, Jul 24 2023

Extensions

More terms from Erich Friedman

New definition from N. J. A. Sloane, Mar 01 2008