login
T(w,h)/2 is the minimum slope >= 1/2 that can be chosen as orientation of a w X h rectangle such that the lower bound for the maximum number of covered grid points A354704(w,d) can be achieved by a suitable translation of the rectangle, where T(w,h) and A354704 are triangles read by rows. T(w,h) = -1 if no slope satisfying this condition exists.
3

%I #18 Dec 19 2024 11:56:22

%S 1,1,1,1,2,2,1,3,2,2,1,1,2,2,2,1,1,2,2,2,2,1,6,2,2,2,1,6,2,6,2,2,2,2,

%T 2,2,1,1,2,2,2,1,1,2,1,2,1,2,2,2,2,2,2,2,2,2,1,2,2,2,2,6,2,1,2,2,1,3,

%U 2,-1,2,2,3,2,1,2,-1,3,2,1,2,2,2,2,6,2,1,2,2,1,2

%N T(w,h)/2 is the minimum slope >= 1/2 that can be chosen as orientation of a w X h rectangle such that the lower bound for the maximum number of covered grid points A354704(w,d) can be achieved by a suitable translation of the rectangle, where T(w,h) and A354704 are triangles read by rows. T(w,h) = -1 if no slope satisfying this condition exists.

%H Hugo Pfoertner, <a href="/A355244/b355244.txt">Table of n, a(n) for n = 1..210</a>, rows 1..20 of triangle, flattened

%H Hugo Pfoertner, <a href="/A355244/a355244.pdf">Illustrations of T(4,2) = 3, T(7,6) = T(9,6) = T(13,12) = 1</a>.

%H Hugo Pfoertner, <a href="/A355244/a355244_1.pdf">Illustrations of T(12,4) = T(12,11) = -1</a>.

%H Hugo Pfoertner, <a href="/A355244/a355244.gp.txt">PARI program</a>

%e The triangle begins:

%e \ h 1 2 3 4 5 6 7 8 9 10 11 12 13

%e w \ --------------------------------------

%e 1 | 1; | | | | | | | | | | | |

%e 2 | 1, 1; | | | | | | | | | | |

%e 3 | 1, 2, 2; | | | | | | | | | |

%e 4 | 1, 3, 2, 2; | | | | | | | | |

%e 5 | 1, 1, 2, 2, 2; | | | | | | | |

%e 6 | 1, 1, 2, 2, 2, 2; | | | | | | |

%e 7 | 1, 6, 2, 2, 2, 1, 6; | | | | | |

%e 8 | 2, 6, 2, 2, 2, 2, 2, 2; | | | | |

%e 9 | 1, 1, 2, 2, 2, 1, 1, 2, 1; | | | |

%e 10 | 2, 1, 2, 2, 2, 2, 2, 2, 2, 2; | | |

%e 11 | 2, 1, 2, 2, 2, 2, 6, 2, 1, 2, 2; | |

%e 12 | 1, 3, 2,-1, 2, 2, 3, 2, 1, 2,-1, 3; |

%e 13 | 2, 1, 2, 2, 2, 2, 6, 2, 1, 2, 2, 1, 2

%e .

%e The first linked illustration shows examples where 2 slopes lead to the same number of covered grid points, where then the smallest multiple of 1/2 is used as a term in the sequence.

%e The second illustration shows the two examples where it is not possible to cover the maximum number of grid points with a rectangle whose side slope is an integer multiple of 1/2.

%o (PARI) /* See Pfoertner link. The program can be used to validate the given terms by calling it successively with the slope parameter k, starting with k = 1/2, 2/2=1, 3/2, (4/2 = 2 already covered by 1/2 via symmetry), 5/2, 6/2=3 for the desired rectangle size w X h, until the number of grid points given by A354704(w,k) is reached for the first time as a result. If the slope parameter is not specified, the program attempts to approximate A354704(w,k) and determine a location of the rectangle that maximizes the free margin between the peripheral grid points and the perimeter of the rectangle. */

%Y Cf. A354704, A354706.

%Y Cf. A355241 (similar, but with number of covered grid points minimized).

%K tabl,sign

%O 1,5

%A _Hugo Pfoertner_, Jun 29 2022