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

0,2

Comments

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.

Links

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

Formula

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.

Example

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.

Mathematica

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

Prog

(GW-BASIC) 10' a256534 First 2^z-1 terms: 20 z=6: defdbl a: for i=1 to z: for j=0 to 2^(i-1)-1: n=n+1: a(n)=4^i + 3*(2*j)^2: print a(n); : next j: next i: end

Crossrefs

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

Sequence in context: A209979 A294629 A160410 * A352205 A364582 A227434

Adjacent sequences: A256531 A256532 A256533 * A256535 A256536 A256537

Keyword

nonn,tabf

Author

Omar E. Pol, Apr 22 2015

Status

approved