A228371 - OEIS

%I #18 Jul 16 2022 01:04:54

%S 1,1,2,2,1,1,3,4,1,1,2,2,1,1,4,8,1,1,2,2,1,1,3,4,1,1,2,2,1,1,5,16,1,1,

%T 2,2,1,1,3,4,1,1,2,2,1,1,4,8,1,1,2,2,1,1,3,4,1,1,2,2,1,1,6,32,1,1,2,2,

%U 1,1,3,4,1,1,2,2,1,1,4,8,1,1,2,2,1,1,3,4,1,1,2,2,1,1,5,16,1,1,2,2,1,1,3,4,1,1,2,2,1,1,4,8,1,1,2,2,1,1,3,4,1,1,2,2,1,1,7,64

%N First differences of A228370. Also A001511 and A006519 interleaved.

%C Number of toothpicks added at n-th stage to the toothpick structure (related to integer compositions) of A228370.

%C The equivalent sequence for integer partitions is A220517.

%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/To#toothpick">Index entries for sequences related to toothpick sequences</a>

%F a(2n-1) = A001511(n), n >= 1. a(2n) = A006519(n), n >= 1.

%e Illustration of the structure after 32 stages. The diagram represents the 16 compositions of 5. The k-th horizontal line segment has length A001511(k) equals the largest part of the k-th region. The k-th vertical line segment has length A006519(k) equals the number of parts of the k-th region.

%e .      _ _ _ _ _

%e 16     _        |

%e 15     _|_      |

%e 14     _  |     |

%e 13     _|_|_    |

%e 12     _    |   |

%e 11     _|_  |   |

%e 10     _  | |   |

%e 9      _|_|_|_  |

%e 8      _      | |

%e 7      _|_    | |

%e 6      _  |   | |

%e 5      _|_|_  | |

%e 4      _    | | |

%e 3      _|_  | | |

%e 2      _  | | | |

%e 1       | | | | |

%e .

%e Written as an irregular triangle the sequence begins:

%e   1,1;

%e   2,2;

%e   1,1,3,4;

%e   1,1,2,2,1,1,4,8;

%e   1,1,2,2,1,1,3,4,1,1,2,2,1,1,5,16;

%e   1,1,2,2,1,1,3,4,1,1,2,2,1,1,4,8,1,1,2,2,1,1,3,4,1,1,2,2,1,1,6,32;

%e   ...

%o (Python)

%o def A228371(n): return ((m:=(n>>1)+1)&-m).bit_length() if n&1 else (m:=n>>1)&-m # _Chai Wah Wu_, Jul 14 2022

%Y Row lengths give 2*A011782. Right border gives A000079.

%Y Cf. A001511, A006519, A139250, A139251, A206437, A220517, A228370.

%K nonn,tabf

%O 1,3

%A _Omar E. Pol_, Aug 21 2013