

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.


0



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
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OFFSET

1,2


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 bottomleft vertex (0,0). A tile is considered crossed if its interior intersects the line segment (0,0)  (n,k).


LINKS



EXAMPLE

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


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


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



