%I #87 Nov 15 2023 15:07:13

%S 0,1,9,21,49,61,97,157,225,237,273,333,417,525,657,813,961,973,1009,

%T 1069,1153,1261,1393,1549,1729,1933,2161,2413,2689,2989,3313,3661,

%U 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

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

%C 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.

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

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

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

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

%H Paolo Xausa, <a href="/A256530/b256530.txt">Table of n, a(n) for n = 0..8191</a>

%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) = (2^i-1)^2 + 3*(2*j)^2: next j: next i

%e With the positive terms written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:

%e 1;

%e 9;

%e 21, 49;

%e 61, 97, 157, 225;

%e 237, 273, 333, 417, 525, 657, 813, 961;

%e ...

%e Right border gives A060867.

%e 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.

%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 After 10 generations there are 273 ON cells, so a(10) = 273.

%t 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_ *)

%o (GW-BASIC) 10' a256530 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)=(2^i-1)^2 + 3*(2*j)^2: print a(n);: next j: next i: end

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

%K nonn,tabf

%O 0,3

%A _Omar E. Pol_, Apr 21 2015