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%I #54 Nov 15 2023 15:06:45
%S 0,4,16,28,64,76,112,172,256,268,304,364,448,556,688,844,1024,1036,
%T 1072,1132,1216,1324,1456,1612,1792,1996,2224,2476,2752,3052,3376,
%U 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 Number of ON cells at n-th stage in simple 2-dimensional cellular automaton (see Comments lines for definition).
%C On the infinite square grid at stage 0 there are no ON cells, so a(0) = 0.
%C At stage 1, four cells are turned ON forming a square, so a(1) = 4.
%C If n is a power of 2 so the structure is a square of side length 2n that contains (2n)^2 ON cells.
%C 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 Has the same rules as A256530 but here a(1) = 4 not 1.
%C Has a smoother behavior than A160410 with which shares infinitely many terms (see example).
%C A261695, the first differences, gives the number of cells turned ON at n-th stage.
%H Michael De Vlieger, <a href="/A256534/b256534.txt">Table of n, a(n) for n = 0..16384</a>
%H 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 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 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%F 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 It appears that a(n) = 4 * A236305(n-1), n >= 1.
%e With the positive terms written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
%e 4;
%e 16;
%e 28, 64;
%e 76, 112, 172, 256;
%e 268, 304, 364, 448, 556, 688, 844, 1024;
%e ...
%e Right border gives the elements of A000302 greater than 1.
%e 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 .
%e Illustration of initial terms, for n = 1..10:
%e . _ _ _ _ _ _ _ _
%e . | _ _ | | _ _ |
%e . | | _|_|_ _ _ _ _ _ _ _ _ _ _ _|_|_ | |
%e . | |_| _ _ _ _ _ _ _ _ _ _ _ _ |_| |
%e . |_ _| | _ _ _ _ | | _ _ _ _ | |_ _|
%e . | | | _ _ | | | | _ _ | | |
%e . | | | | _|_|_|_ _|_|_|_ | | | |
%e . | | | |_| _ _ _ _ |_| | | |
%e . | | |_ _| | _|_ _|_ | |_ _| | |
%e . | |_ _ _| |_| _ _ |_| |_ _ _| |
%e . | | | | | | | |
%e . | _ _ _| _| |_ _| |_ |_ _ _ |
%e . | | _ _| | |_ _ _ _| | |_ _ | |
%e . | | | _| |_ _| |_ _| |_ | | |
%e . | | | | |_ _ _ _ _ _ _ _| | | | |
%e . | | | |_ _| | | | | |_ _| | | |
%e . _ _| | |_ _ _ _| | | |_ _ _ _| | |_ _
%e . | _| |_ _ _ _ _ _| |_ _ _ _ _ _| |_ |
%e . | | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
%e . | |_ _| | | |_ _| |
%e . |_ _ _ _| |_ _ _ _|
%e .
%e After 10 generations there are 304 ON cells, so a(10) = 304.
%t {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 (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 Cf. A000302, A011782, A139250, A147562, A160410, A160414, A236305, A256530, A261695.
%K nonn,tabf
%O 0,2
%A _Omar E. Pol_, Apr 22 2015
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