A256264 - OEIS

%I #30 Feb 14 2017 15:16:34

%S 0,1,2,5,6,9,14,21,22,25,30,37,42,53,70,85,86,89,94,101,106,117,134,

%T 149,154,165,182,205,234,269,310,341,342,345,350,357,362,373,390,405,

%U 410,421,438,461,490,525,566,597,602,613,630,653,682,717,758,805,858,917,982,1053,1130,1213,1302,1365

%N Partial sums of A256263.

%C First differs from A255747 at a(27).

%H Ivan Neretin, <a href="/A256264/b256264.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 a(n) = (A256260(n+1) - 1)/4.

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

%e 0,

%e 1,

%e 2,   5,

%e 6,   9, 14,  21,

%e 22, 25, 30,  37,  42,  53,  70,  85;

%e 86, 89, 94, 101, 106, 117, 134, 149, 154, 165, 182, 205, 234, 269,310,341;

%e ...

%e It appears that the first column gives 0 together with the terms of A047849, hence the right border gives A002450.

%e It appears that this triangle at least shares with the triangles from the following sequences; A151920, A255737, A255747, A256249, the positive elements of the columns k, if k is a power of 2.

%e From _Omar E. Pol, Jan 02 2016: (Start)

%e Illustration of initial terms in the fourth quadrant of the square grid:

%e ---------------------------------------------------------------------------

%e n    a(n)                 Compact diagram

%e ---------------------------------------------------------------------------

%e 0     0     _

%e 1     1    |_|_ _

%e 2     2      |_| |

%e 3     5      |_ _|_ _ _ _

%e 4     6          |_| | | |

%e 5     9          |_ _| | |

%e 6    14          |_ _ _| |

%e 7    21          |_ _ _ _|_ _ _ _ _ _ _ _

%e 8    22                  |_| | | |_ _  | |

%e 9    25                  |_ _| | |_  | | |

%e 10   30                  |_ _ _| | | | | |

%e 11   37                  |_ _ _ _| | | | |

%e 12   42                  | | |_ _ _| | | |

%e 13   53                  | |_ _ _ _ _| | |

%e 14   70                  |_ _ _ _ _ _ _| |

%e 15   85                  |_ _ _ _ _ _ _ _|_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

%e 16   86                                  |_| | | |_ _  | |_ _ _ _ _ _  | |

%e 17   89                                  |_ _| | |_  | | |_ _ _ _ _  | | |

%e 18   94                                  |_ _ _| | | | | |_ _ _ _  | | | |

%e 19  101                                  |_ _ _ _| | | | |_ _ _  | | | | |

%e 20  106                                  | | |_ _ _| | | |_ _  | | | | | |

%e 21  117                                  | |_ _ _ _ _| | |_  | | | | | | |

%e 22  134                                  |_ _ _ _ _ _ _| | | | | | | | | |

%e 23  149                                  |_ _ _ _ _ _ _ _| | | | | | | | |

%e 24  154                                  | | | | | | |_ _ _| | | | | | | |

%e 25  165                                  | | | | | |_ _ _ _ _| | | | | | |

%e 26  182                                  | | | | |_ _ _ _ _ _ _| | | | | |

%e 27  205                                  | | | |_ _ _ _ _ _ _ _ _| | | | |

%e 28  234                                  | | |_ _ _ _ _ _ _ _ _ _ _| | | |

%e 29  269                                  | |_ _ _ _ _ _ _ _ _ _ _ _ _| | |

%e 30  310                                  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |

%e 31  341                                  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|

%e .

%e a(n) is also the total number of cells in the first n regions of the diagram. A256263(n) gives the number of cells in the n-th region of the diagram.

%e (End)

%t Accumulate@Flatten@Join[{0}, NestList[Join[#, Range[Length[#] - 1]*6 - 1, {2 #[[-1]] + 1}] &, {1}, 5]] (* _Ivan Neretin_, Feb 14 2017 *)

%Y Cf. A002450, A011782, A047849, A139250, A151920, A255737, A255747, A256249, A256258, A256260, A256261, A256263, A256265.

%K nonn,look

%O 0,3

%A _Omar E. Pol_, Mar 30 2015