A159913 a(n) = 2^(A000120(n) + 1) - 1, where A000120(n) = number of nonzero bits in n.

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, 3, 7, 7, 15, 7, 15, 15, 31, 7, 15, 15, 31, 15, 31, 31, 63, 7, 15, 15, 31, 15, 31, 31, 63, 15, 31, 31, 63, 31, 63, 63, 127, 3, 7, 7, 15, 7, 15, 15, 31, 7, 15, 15, 31, 15, 31, 31

Offset: 0

Author

M. F. Hasler, May 03 2009

Keywords

• nonn

Comments

Essentially the same sequence as A117973 and A001316. The latter entry has much more information. - N. J. A. Sloane, Jun 05 2009

First differences of A159912; every other term of A038573.

Equals Sierpinski's gasket, A047999; as an infinite lower triangular matrix * [1,2,2,2,...] as a vector. - Gary W. Adamson, Oct 16 2009

a(n) is also the number of cells turned ON at n-th generation in the outward corner version of the Ulam-Warburton cellular automaton of A147562, and a(n) is also the number of Y-toothpicks added at n-th generation in the outward corner version of the Y-toothpick structure of A160120. - David Applegate and Omar E. Pol, Jan 24 2016

Examples

			From _Michael De Vlieger_, Jan 25 2016: (Start)
The number n converted to binary, "0" represented by "." for better visibility of 1's, totaling the 1's and calculating the sequence:
n    Binary   Total                         a(n)
0 -> .     ->     0, thus 2^(0+1)-1 =  2-1 =  1
1 -> 1     ->     1,   "  2^(1+1)-1 =  4-1 =  3
2 -> 1.    ->     1,   "  2^(1+1)-1 =  4-1 =  3
3 -> 11    ->     2,   "  2^(2+1)-1 =  8-1 =  7
4 -> 1..   ->     1,   "  2^(1+1)-1 =  4-1 =  3
5 -> 1.1   ->     2,   "  2^(2+1)-1 =  8-1 =  7
6 -> 11.   ->     2,   "  2^(2+1)-1 =  8-1 =  7
7 -> 111   ->     3,   "  2^(3+1)-1 = 16-1 = 15
8 -> 1...  ->     1,   "  2^(1+1)-1 =  4-1 =  3
9 -> 1..1  ->     2,   "  2^(2+1)-1 =  8-1 =  7
10-> 1.1.  ->     2,   "  2^(2+1)-1 =  8-1 =  7
(End)
		

Links

• David Applegate, The movie version
• 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

Crossrefs

Rows of triangle in A038573 converge to this sequence. - N. J. A. Sloane, Jun 05 2009

Cf. A000120, A038573, A047999, A159912, A117973, A001316, A147582, A160121.

Programs

Mathematica

Table[2^(DigitCount[n, 2][[1]] + 1) - 1, {n, 0, 78}] (* or *)
Table[2^(Total@ IntegerDigits[n, 2] + 1) - 1, {n, 0, 78}] (* Michael De Vlieger, Jan 25 2016 *)

PARI

A159913(n)=2<
				

Python

def A159913(n): return (1<Chai Wah Wu, Nov 15 2022

Formula

a(n) = 2^A000120(2n+1) - 1 = A038573(2n+1) = 2*A038573(n) + 1 = A159912(n+1) - A159912(n).

a(n) = A160019(n,n). - Philippe Deléham, Nov 15 2011

a(n) = n - Sum_{k=0..n} (-1)^binomial(n, k). - Peter Luschny, Jan 14 2018