

A255748


Total number of ON states after n generations of cellular automaton based on triangles in a 60degree wedge (see Comments lines for definition).


4



1, 3, 4, 8, 11, 13, 14, 22, 29, 35, 40, 44, 47, 49, 50, 66, 81, 95, 108, 120, 131, 141, 150, 158, 165, 171, 176, 180, 183, 185, 186, 218, 249, 279, 308, 336, 363, 389, 414, 438, 461, 483, 504, 524, 543, 561, 578, 594, 609, 623, 636, 648, 659, 669, 678, 686, 693, 699, 704, 708, 711, 713, 714, 778, 841, 903, 964, 1024
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

In order to construct the structure we use the following rules:
On the infinite triangular grid we are in a 60degree wedge with the vertex located on top of the wedge.
The nearest triangular cell to the vertex remains OFF.
At stage 1, we turn ON the cell whose base is adjacent to the previous OFF cell.
At stage n, in the nth level of the structure, we turn ON k cells connected by their vertices with their bases up, where k = A080079(n).
The cells turned ON remain ON forever.
The structure seems to grow into the holes of a virtual Sierpiński's triangle (see example).
Note that this is also the structure in every one of the six wedges of the structure of A256266.
A080079 gives the number of cells turned ON at nth stage.


LINKS



FORMULA



EXAMPLE

Illustration of initial terms:


. / \
1 1 1 / T \
2 2 3 / T T \
3 1 4 / T \
4 4 8 / T T T T \
5 3 11 / T T T \
6 2 13 / T T \
7 1 14 / T \
8 8 22 / T T T T T T T T \
9 7 29 / T T T T T T T \
10 6 35 / T T T T T T \
11 5 40 / T T T T T \
12 4 44 / T T T T \
13 3 47 / T T T \
14 2 49 / T T \
15 1 50 / T \
...
For n = 15 after 15 generations there are 50 ON cells in the structure, so a(15) = 50.


MATHEMATICA

Accumulate@ Flatten@ Table[Range[2^n, 1, 1], {n, 0, 6}] (* Michael De Vlieger, Nov 03 2022 *)


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



