OFFSET
0,2
COMMENTS
Essentially the same sequence as A117973 and A001316. The latter entry has much more information. - N. J. A. Sloane, Jun 05 2009
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
LINKS
FORMULA
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
EXAMPLE
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)
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 *)
PROG
(PARI) A159913(n)=2<<norml2(binary(n))-1
(Python 3.10+)
def A159913(n): return (1<<n.bit_count()+1)-1 # Chai Wah Wu, Nov 15 2022
CROSSREFS
Rows of triangle in A038573 converge to this sequence. - N. J. A. Sloane, Jun 05 2009
KEYWORD
nonn
AUTHOR
M. F. Hasler, May 03 2009
STATUS
approved