OFFSET
0,2
COMMENTS
Partial sums give A233774.
The subsequence of odd terms is A083318. - Gary W. Adamson, Jan 13 2014
Equivalently, this is the coordination sequence for the Sierpinski gasket with respect to the apex. - N. J. A. Sloane, Sep 19 2020
LINKS
N. J. A. Sloane, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Eric Weisstein's World of Mathematics, Sierpiński Gasket Graph.
Wikipedia, Sierpinski triangle.
FORMULA
a(0)=1, a(n) = (2^t(n) + 1) * 2^(c(n) - 1) where t(n) = A007814(n) is the number of trailing zeros in the binary representation of n and c(n) = A000120(n) is the total number of ones in the binary representation of n. - Johan Falk, Jun 24 2020
EXAMPLE
Illustration of initial terms:
--------------------------------------------------------
Diagram n a(n) A233774(n)
--------------------------------------------------------
* 0 1 1
/T\
*---* 1 2 3
/T\ /T\
*---*---* 2 3 6
/T\ /T\
*---* *---* 3 4 10
/T\ /T\ /T\ /T\
*---*---*---*---* 4 5 15
/T\ /T\
*---* *---* 5 4 19
--------------------------------------------------------
After five stages the number of "black" triangles in the structure is A006046(5) = 11 and the number of "black" triangles in row 5 is A001316(5-1) = 2. The number of vertices in row 5 is equal to 4, so a(5) = 4.
Written as an irregular triangle the sequence begins:
1;
2;
3;
4,5;
4,6,8,9;
4,6,8,10,8,12,16,17;
4,6,8,10,8,12,16,18,8,12,16,20,16,24,32,33;
...
MAPLE
MATHEMATICA
A233775[n_] := If[n == 0, 1, (2^IntegerExponent[n, 2]+1)*2^(DigitSum[n, 2]-1)];
Array[A233775, 100, 0] (* Paolo Xausa, Aug 05 2024 *)
PROG
(PARI) print1("1, "); for(k=1, 70, print1((2^valuation(k, 2)+1) *2^(hammingweight(k)-1), ", ")) \\ Hugo Pfoertner, Jul 27 2020
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Dec 16 2013
STATUS
approved