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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 (list; graph; refs; listen; history; text; internal format)
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.
LINKS
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 37.
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 *)
CROSSREFS
Sequence in context: A243177 A113294 A169691 * A152412 A287404 A287388
KEYWORD
nonn,look
AUTHOR
Omar E. Pol, Mar 30 2015
STATUS
approved

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Last modified February 25 19:44 EST 2024. Contains 370332 sequences. (Running on oeis4.)