A256530 Number of ON cells at n-th stage in simple 2-dimensional cellular automaton (see Comments lines for definition).

0, 1, 9, 21, 49, 61, 97, 157, 225, 237, 273, 333, 417, 525, 657, 813, 961, 973, 1009, 1069, 1153, 1261, 1393, 1549, 1729, 1933, 2161, 2413, 2689, 2989, 3313, 3661, 3969, 3981, 4017, 4077, 4161, 4269, 4401, 4557, 4737, 4941, 5169, 5421, 5697, 5997, 6321, 6669, 7041, 7437, 7857, 8301, 8769, 9261, 9777, 10317, 10881, 11469

Offset: 0

Author

Omar E. Pol, Apr 21 2015

Keywords

• nonn
• tabf

Comments

On the infinite square grid at stage 0 there are no ON cells, so a(0) = 0.

At stage 1, only one cell is turned ON, so a(1) = 1.

If n is a power of 2 so the structure is a square of side length 2n - 1 that contains (2n-1)^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).

Note that a(24) = 1729 is also the Hardy-Ramanujan number (see A001235).

Has the same rules as A256534 but here a(1) = 1 not 4.

Has a smoother behavior than A160414 with which shares infinitely many terms (see example).

A256531, the first differences, gives the number of cells turned ON at n-th stage.

Examples

			With the positive terms written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
1;
9;
21,    49;
61,    97,  157,  225;
237,  273,  333,  417,  525,  657,  813,  961;
...
Right border gives A060867.
This triangle T(n,k) shares with the triangle A160414 the terms of the column k, if k is a power of 2, for example both triangles share the following terms: 1, 9, 21, 49, 61, 97, 225, 237, 273, 417, 961, etc.
.
Illustration of initial terms, for n = 1..10:
.       _ _ _ _                       _ _ _ _
.      |  _ _  |                     |  _ _  |
.      | |  _|_|_ _ _ _ _ _ _ _ _ _ _|_|_  | |
.      | |_|  _ _ _ _ _ _   _ _ _ _ _ _  |_| |
.      |_ _| |  _ _ _ _  | |  _ _ _ _  | |_ _|
.          | | |  _ _  | | | |  _ _  | | |
.          | | | |  _|_|_|_|_|_|_  | | | |
.          | | | |_|  _ _   _ _  |_| | | |
.          | | |_ _| |  _|_|_  | |_ _| | |
.          | |_ _ _| |_|  _  |_| |_ _ _| |
.          |  _ _ _|  _| |_| |_  |_ _ _  |
.          | |  _ _| | |_ _ _| | |_ _  | |
.          | | |  _| |_ _| |_ _| |_  | | |
.          | | | | |_ _ _ _ _ _ _| | | | |
.          | | | |_ _| | | | | |_ _| | | |
.       _ _| | |_ _ _ _| | | |_ _ _ _| | |_ _
.      |  _| |_ _ _ _ _ _| |_ _ _ _ _ _| |_  |
.      | | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
.      | |_ _| |                     | |_ _| |
.      |_ _ _ _|                     |_ _ _ _|
.
After 10 generations there are 273 ON cells, so a(10) = 273.
		

Links

• 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

Crossrefs

Cf. A000225, A011782, A001235, A060867, A139250, A147562, A160410, A160414, A256531, A256534.

Programs

Mathematica

With[{z=7},Join[{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 *)

Formula

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