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

%I A256534 #54 Nov 15 2023 15:06:45
%S A256534 0,4,16,28,64,76,112,172,256,268,304,364,448,556,688,844,1024,1036,
%T A256534 1072,1132,1216,1324,1456,1612,1792,1996,2224,2476,2752,3052,3376,
%U A256534 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
%N A256534 Number of ON cells at n-th stage in simple 2-dimensional cellular automaton (see Comments lines for definition).
%C A256534 On the infinite square grid at stage 0 there are no ON cells, so a(0) = 0.
%C A256534 At stage 1, four cells are turned ON forming a square, so a(1) = 4.
%C A256534 If n is a power of 2 so the structure is a square of side length 2n that contains (2n)^2 ON cells.
%C A256534 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).
%C A256534 Has the same rules as A256530 but here a(1) = 4 not 1.
%C A256534 Has a smoother behavior than A160410 with which shares infinitely many terms (see example).
%C A256534 A261695, the first differences, gives the number of cells turned ON at n-th stage.
%H A256534 Michael De Vlieger, <a href="/A256534/b256534.txt">Table of n, a(n) for n = 0..16384</a>
%H A256534 Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, <a href="https://arxiv.org/abs/2210.10968">Identities and periodic oscillations of divide-and-conquer recurrences splitting at half</a>, arXiv:2210.10968 [cs.DS], 2022, p. 37.
%H A256534 N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%H A256534 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%F A256534 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
%F A256534 It appears that a(n) = 4 * A236305(n-1), n >= 1.
%e A256534 With the positive terms written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
%e A256534 4;
%e A256534 16;
%e A256534 28,     64;
%e A256534 76,    112,  172,  256;
%e A256534 268,   304,  364,  448,  556,  688,  844, 1024;
%e A256534 ...
%e A256534 Right border gives the elements of A000302 greater than 1.
%e A256534 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.
%e A256534 .
%e A256534 Illustration of initial terms, for n = 1..10:
%e A256534 .       _ _ _ _                         _ _ _ _
%e A256534 .      |  _ _  |                       |  _ _  |
%e A256534 .      | |  _|_|_ _ _ _ _ _ _ _ _ _ _ _|_|_  | |
%e A256534 .      | |_|  _ _ _ _ _ _     _ _ _ _ _ _  |_| |
%e A256534 .      |_ _| |  _ _ _ _  |   |  _ _ _ _  | |_ _|
%e A256534 .          | | |  _ _  | |   | |  _ _  | | |
%e A256534 .          | | | |  _|_|_|_ _|_|_|_  | | | |
%e A256534 .          | | | |_|  _ _     _ _  |_| | | |
%e A256534 .          | | |_ _| |  _|_ _|_  | |_ _| | |
%e A256534 .          | |_ _ _| |_|  _ _  |_| |_ _ _| |
%e A256534 .          |       |   | |   | |   |       |
%e A256534 .          |  _ _ _|  _| |_ _| |_  |_ _ _  |
%e A256534 .          | |  _ _| | |_ _ _ _| | |_ _  | |
%e A256534 .          | | |  _| |_ _|   |_ _| |_  | | |
%e A256534 .          | | | | |_ _ _ _ _ _ _ _| | | | |
%e A256534 .          | | | |_ _| | |   | | |_ _| | | |
%e A256534 .       _ _| | |_ _ _ _| |   | |_ _ _ _| | |_ _
%e A256534 .      |  _| |_ _ _ _ _ _|   |_ _ _ _ _ _| |_  |
%e A256534 .      | | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
%e A256534 .      | |_ _| |                       | |_ _| |
%e A256534 .      |_ _ _ _|                       |_ _ _ _|
%e A256534 .
%e A256534 After 10 generations there are 304 ON cells, so a(10) = 304.
%t A256534 {0}~Join~Flatten@ Table[4^i + 3 (2 j)^2, {i, 6}, {j, 0, 2^(i - 1) - 1}] (* _Michael De Vlieger_, Nov 03 2022 *)
%o A256534 (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
%Y A256534 Cf. A000302, A011782, A139250, A147562, A160410, A160414, A236305, A256530, A261695.
%K A256534 nonn,tabf
%O A256534 0,2
%A A256534 _Omar E. Pol_, Apr 22 2015