A256534 - OEIS

%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