A292421 Square array T(n,k) = number of tiles crossed by the line segment (0,0) -- (n,k) in a running bond pattern tiling with square tiles, read by antidiagonals.

1, 2, 2, 4, 3, 3, 4, 3, 4, 4, 5, 4, 4, 5, 5, 6, 8, 6, 7, 6, 6, 8, 7, 5, 6, 6, 7, 7, 8, 9, 6, 8, 8, 7, 8, 8, 9, 8, 12, 9, 7, 8, 10, 9, 9, 10, 10, 10, 7, 10, 10, 10, 9, 10, 10, 12, 11, 11, 8, 10, 9, 13, 10, 10, 11, 11, 12, 11, 12, 16, 12, 11, 12, 10, 12, 13, 12, 12

Offset: 1

Author

Luc Rousseau, Sep 16 2017

Keywords

• nonn
• tabl

Comments

Running bond pattern: the tiles form horizontal rows and for all i (row number), row i+1 is row i translated by vector (1/2, 1). The first row is supposed to contain the tile with bottom-left vertex (0,0). A tile is considered crossed if its interior intersects the line segment (0,0) -- (n,k).

Examples

			T(3,5) = 5 because (0,0) -- (3,5) crosses the following tiles, identified by their bottom-left vertices: (0,0), (0.5,1), (1,2), (1.5,3), (2,4).
T(5,3) = 6 because (0,0) -- (5,3) crosses the following tiles, identified by their bottom-left vertices: (0,0), (1,0), (1.5,1), (2.5,1), (3,2), (4,2).
		

Crossrefs

Cf. A074712.

Programs

Mathematica

F[a_, b_, p_, q_, i_] :=
  Block[{x0, x1, d}, x0 = (p/q - a/b)*i; x1 = x0 + p/q;
   d = Floor[x1] - Floor[x0]; If[IntegerQ[x1], d, d + 1]];
FF[a_, b_, p_, q_] := Sum[F[a, b, p, q, i], {i, 0, q - 1}];
a = 1; b = 2;
Table[FF[a, b, p, s - p], {s, 2, 13}, {p, 1, s - 1}] // Flatten