A261695 -id:A261695 - cp's OEIS frontend

A256534 Number of ON cells at n-th stage in simple 2-dimensional cellular automaton (see Comments lines for definition).

0, 4, 16, 28, 64, 76, 112, 172, 256, 268, 304, 364, 448, 556, 688, 844, 1024, 1036, 1072, 1132, 1216, 1324, 1456, 1612, 1792, 1996, 2224, 2476, 2752, 3052, 3376, 3724, 4096, 4108, 4144, 4204, 4288, 4396, 4528, 4684, 4864, 5068, 5296, 5548, 5824, 6124, 6448, 6796, 7168, 7564, 7984, 8428, 8896, 9388, 9904, 10444, 11008

Offset: 0

Omar E. Pol, Apr 22 2015

On the infinite square grid at stage 0 there are no ON cells, so a(0) = 0.
At stage 1, four cells are turned ON forming a square, so a(1) = 4.
If n is a power of 2 so the structure is a square of side length 2n that contains (2n)^2 ON cells.
The structure grows by the four corners as square waves forming layers of ON cells up the next square structure, and so on (see example).
Has the same rules as A256530 but here a(1) = 4 not 1.
Has a smoother behavior than A160410 with which shares infinitely many terms (see example).
A261695, the first differences, gives the number of cells turned ON at n-th stage.

			With the positive terms written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
4;
16;
28,     64;
76,    112,  172,  256;
268,   304,  364,  448,  556,  688,  844, 1024;
...
Right border gives the elements of A000302 greater than 1.
This triangle T(n,k) shares with the triangle A160410 the terms of the column k, if k is a power of 2, for example, both triangles share the following terms: 4, 16, 28, 64, 76, 112, 256, 268, 304, 448, 1024, etc.
.
Illustration of initial terms, for n = 1..10:
.       _ _ _ _                         _ _ _ _
.      |  _ _  |                       |  _ _  |
.      | |  _|_|_ _ _ _ _ _ _ _ _ _ _ _|_|_  | |
.      | |_|  _ _ _ _ _ _     _ _ _ _ _ _  |_| |
.      |_ _| |  _ _ _ _  |   |  _ _ _ _  | |_ _|
.          | | |  _ _  | |   | |  _ _  | | |
.          | | | |  _|_|_|_ _|_|_|_  | | | |
.          | | | |_|  _ _     _ _  |_| | | |
.          | | |_ _| |  _|_ _|_  | |_ _| | |
.          | |_ _ _| |_|  _ _  |_| |_ _ _| |
.          |       |   | |   | |   |       |
.          |  _ _ _|  _| |_ _| |_  |_ _ _  |
.          | |  _ _| | |_ _ _ _| | |_ _  | |
.          | | |  _| |_ _|   |_ _| |_  | | |
.          | | | | |_ _ _ _ _ _ _ _| | | | |
.          | | | |_ _| | |   | | |_ _| | | |
.       _ _| | |_ _ _ _| |   | |_ _ _ _| | |_ _
.      |  _| |_ _ _ _ _ _|   |_ _ _ _ _ _| |_  |
.      | | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
.      | |_ _| |                       | |_ _| |
.      |_ _ _ _|                       |_ _ _ _|
.
After 10 generations there are 304 ON cells, so a(10) = 304.

Michael De Vlieger, Table of n, a(n) for n = 0..16384
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.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A000302, A011782, A139250, A147562, A160410, A160414, A236305, A256530, A261695.

{0}~Join~Flatten@ Table[4^i + 3 (2 j)^2, {i, 6}, {j, 0, 2^(i - 1) - 1}] (* Michael De Vlieger, Nov 03 2022 *)

For i = 1 to z: for j = 0 to 2^(i-1)-1: n = n+1: a(n) = 4^i + 3*(2*j)^2: next j: next i

It appears that a(n) = 4 * A236305(n-1), n >= 1.

A256531 First differences of A256530.

Original entry on oeis.org

0, 1, 8, 12, 28, 12, 36, 60, 68, 12, 36, 60, 84, 108, 132, 156, 148, 12, 36, 60, 84, 108, 132, 156, 180, 204, 228, 252, 276, 300, 324, 348, 308, 12, 36, 60, 84, 108, 132, 156, 180, 204, 228, 252, 276, 300, 324, 348, 372, 396, 420, 444, 468, 492, 516, 540, 564, 588, 612, 636, 660, 684, 708, 732, 628, 12, 36, 60, 84, 108

Offset: 0

Omar E. Pol, Apr 21 2015

Number of cells turned ON at n-th stage of cellular automaton of A256530.

Similar to A261695 which shares infinitely many terms.

			With the positive terms written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
1;
8;
12, 28;
12, 36, 60, 68;
12, 36, 60, 84, 108, 132, 156, 148;
12, 36, 60, 84, 108, 132, 156, 180, 204, 228, 252, 276, 300, 324, 348, 308;
...
The terms of the rows that start with 12 are also the initial terms of A073762, except the last term of every row, hence rows converge to A073762.

Paolo Xausa, Table of n, a(n) for n = 0..8191
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A011782, A073762, A139251, A147582, A161411, A161415, A256530, A261695.

With[{z=7},Differences[Join[{0,0},Flatten[Array[(2^#-1)^2+12Range[0,2^(#-1)-1]^2&,z]]]]] (* Generates 2^z terms *) (* Paolo Xausa, Nov 15 2023, after Omar E. Pol *)

cp's OEIS Frontend

A256534 Number of ON cells at n-th stage in simple 2-dimensional cellular automaton (see Comments lines for definition).

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