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A255748
Total number of ON states after n generations of cellular automaton based on triangles in a 60-degree 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
OFFSET
1,2
COMMENTS
Also partial sums of A080079.
In order to construct the structure we use the following rules:
On the infinite triangular grid we are in a 60-degree 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 n-th 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 n-th stage.
FORMULA
a(n) = A256266(n)/6.
EXAMPLE
Illustration of initial terms:
-----------------------------------------------------------
n A080079(n) a(n) Diagram
-----------------------------------------------------------
. / \
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 *)
KEYWORD
nonn,look
AUTHOR
Omar E. Pol, Mar 30 2015
STATUS
approved