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 upper bound for the minimum number of covered grid points A354702(w,d) can be achieved by a suitable translation of the rectangle, where T(w,h) and A354702 are triangles read by rows. T(w,h) = -1 if no slope satisfying this condition exists.
4

%I #16 Aug 02 2024 11:38:37

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

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

%U 1,2,6,2,2,1,2,2,2,2,2,2,1,2,6,2,2,1,2,2,2,2,2

%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 upper bound for the minimum number of covered grid points A354702(w,d) can be achieved by a suitable translation of the rectangle, where T(w,h) and A354702 are triangles read by rows. T(w,h) = -1 if no slope satisfying this condition exists.

%C No example of T(w,h) = -1 is known for w <= 20, i.e., the upper bound A354702(w,h) can always be achieved using a slope that is an integer multiple of 1/2. In the range w <= 20, T(17,13) = 3 is the only occurrence of the required slope 3/2.

%C For some rectangle dimensions it is possible to reach the value of A354702(w,h) with different slopes. In the simplest case, e.g., with the slopes 1/2 (T(w,h)=1) and 1 (A355242(w,h)=1). The linked file shows examples for some pairs of values (w,h) and the case of (10,10) with 3 different slopes.

%H Hugo Pfoertner, <a href="/A355241/b355241.txt">Table of n, a(n) for n = 1..210</a>, rows 1..20 of triangle, flattened

%H Hugo Pfoertner, <a href="/A355241/a355241.pdf">Different slopes with the same number of grid points covered</a>.

%H Hugo Pfoertner, <a href="/A355241/a355241.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, 2; | | | | | | | | | | |

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

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

%e 5 | 2, 2, 1, 1, 6; | | | | | | | |

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

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

%e 8 | 2, 2, 1, 1, 6, 1, 2, 1; | | | | |

%e 9 | 2, 2, 1, 2, 6, 2, 2, 2, 2; | | | |

%e 10 | 2, 2, 1, 1, 6, 6, 2, 1, 2, 1; | | |

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

%e 12 | 2, 2, 1, 2, 6, 2, 2, 1, 2, 2, 2, 2; |

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

%o (PARI) /* see 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 A354702(w,k) is reached for the first time as a result. Without specifying the slope parameter, the program tries to approximate A354702(w,k) and determine a position of the rectangle maximizing the free space between peripheral grid points and the rectangle. */

%Y Cf. A354702, A355242.

%Y A355244 is similar, but for maximizing the number of covered grid points.

%K nonn,tabl

%O 1,3

%A _Hugo Pfoertner_, Jun 27 2022