This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A355241 #18 Dec 19 2024 11:57:12 %S A355241 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 A355241 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 A355241 1,2,6,2,2,1,2,2,2,2,2,2,1,2,6,2,2,1,2,2,2,2,2 %N A355241 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 A355241 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 A355241 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 A355241 Hugo Pfoertner, <a href="/A355241/b355241.txt">Table of n, a(n) for n = 1..210</a>, rows 1..20 of triangle, flattened %H A355241 Hugo Pfoertner, <a href="/A355241/a355241.pdf">Different slopes with the same number of grid points covered</a>. %H A355241 Hugo Pfoertner, <a href="/A355241/a355241.gp.txt">PARI program</a> %e A355241 The triangle begins: %e A355241 \ h 1 2 3 4 5 6 7 8 9 10 11 12 13 %e A355241 w \ -------------------------------------- %e A355241 1 | 1; | | | | | | | | | | | | %e A355241 2 | 1, 2; | | | | | | | | | | | %e A355241 3 | 1, 1, 1; | | | | | | | | | | %e A355241 4 | 2, 2, 1, 1; | | | | | | | | | %e A355241 5 | 2, 2, 1, 1, 6; | | | | | | | | %e A355241 6 | 2, 2, 1, 1, 6, 2; | | | | | | | %e A355241 7 | 2, 2, 1, 2, 2, 2, 2; | | | | | | %e A355241 8 | 2, 2, 1, 1, 6, 1, 2, 1; | | | | | %e A355241 9 | 2, 2, 1, 2, 6, 2, 2, 2, 2; | | | | %e A355241 10 | 2, 2, 1, 1, 6, 6, 2, 1, 2, 1; | | | %e A355241 11 | 2, 2, 1, 2, 6, 2, 2, 1, 2, 1, 2; | | %e A355241 12 | 2, 2, 1, 2, 6, 2, 2, 1, 2, 2, 2, 2; | %e A355241 13 | 2, 2, 1, 2, 6, 2, 2, 1, 2, 2, 2, 2, 2 %o A355241 (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 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 A355241 Cf. A354702, A355242. %Y A355241 A355244 is similar, but for maximizing the number of covered grid points. %K A355241 nonn,tabl %O A355241 1,3 %A A355241 _Hugo Pfoertner_, Jun 27 2022