A348149 - OEIS

%I #25 Jun 06 2023 03:09:00

%S 4,9,14,20,26,33,40,48,55,64

%N Variation of the Barnyard sequence A347581: a(n) is the minimum number of unit-length line segments required to enclose areas of 1 through n on a square grid when the number of segments is minimized as each area of incrementing size, starting at 1, is added.

%C In this variation of A347581 the areas must be added in the order of their sizes, from 1 through n, and as each area is added the minimum possible number of line segments must be used. This forces, for example, the first three areas of size 1, 2 and 3 to form a 2 X 3 block and thus they can never appear in any other arrangement in the final area. This is also true for n up to at least 9 due to the restriction of maximizing the usable edges for the next area. This leads to a(8) and a(10) containing one more line segment than the optimal solutions of A347581.

%H Sascha Kurz, <a href="https://arxiv.org/abs/math/0506428">Counting polyominoes with minimum perimeter</a>, arXiv:math/0506428 [math.CO], 2015.

%e Examples of n = 1 to n = 10 are given below. Note that for a(3) the configuration could also consist of the area of size 1 sitting above the area of size 2 with the area of size 3 forming an L-shaped block creating the minimal 2 X 3 block.

%e .

%e    __

%e   |__|  a(1) = 4

%e    __ __ __

%e   |__|__ __|  a(2) = 9

%e    __ __ __

%e   |__|__ __|  a(3) = 14

%e   |__ __ __|

%e    __ __ __

%e   |__|__ __|

%e   |__ __ __|  a(4) = 20

%e   |     |

%e   |__ __|

%e    __ __ __

%e   |__|__ __|__

%e   |__ __ __|  |  a(5) = 26

%e   |     |     |

%e   |__ __|__ __|

%e    __ __ __

%e   |__|__ __|__ __ __

%e   |__ __ __|  |     |  a(6) = 33

%e   |     |     |     |

%e   |__ __|__ __|__ __|

%e          __ __ __ __

%e    __ __|__         |

%e   |__|__ __|__ __ __|

%e   |__ __ __|  |     |  a(7) = 40

%e   |     |     |     |

%e   |__ __|__ __|__ __|

%e          __ __ __ __

%e         |           |

%e         |__ __ __ __|

%e    __ __|__         |

%e   |__|__ __|__ __ __|  a(8) = 48

%e   |__ __ __|  |     |

%e   |     |     |     |

%e   |__ __|__ __|__ __|

%e    __ __ __ __ __ __ __

%e   |        |           |

%e   |        |__ __ __ __|

%e   |__ __ __|__         |

%e      |__|__ __|__ __ __|  a(9) = 55

%e      |__ __ __|  |     |

%e      |     |     |     |

%e      |__ __|__ __|__ __|

%e       __ __ __ __ __ __ __

%e      |        |           |

%e      |        |__ __ __ __|

%e    __|__ __ __|__         |

%e   |     |__|__ __|__ __ __|  a(10) = 64

%e   |     |__ __ __|  |     |

%e   |     |     |     |     |

%e   |     |__ __|__ __|__ __|

%e   |__ __|

%e .

%Y Cf. A347581, A001168, A291808, A291809, A328020, A291806, A006983.

%K nonn,more

%O 1,1

%A _Scott R. Shannon_, Oct 03 2021