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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.

%I A292421 #20 Sep 18 2017 12:54:23
%S A292421 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,
%T A292421 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,
%U A292421 13,10,10,11,11,12,11,12,16,12,11,12,10,12,13,12,12
%N 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.
%C A292421 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).
%e A292421 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).
%e A292421 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).
%t A292421 F[a_, b_, p_, q_, i_] :=
%t A292421   Block[{x0, x1, d}, x0 = (p/q - a/b)*i; x1 = x0 + p/q;
%t A292421    d = Floor[x1] - Floor[x0]; If[IntegerQ[x1], d, d + 1]];
%t A292421 FF[a_, b_, p_, q_] := Sum[F[a, b, p, q, i], {i, 0, q - 1}];
%t A292421 a = 1; b = 2;
%t A292421 Table[FF[a, b, p, s - p], {s, 2, 13}, {p, 1, s - 1}] // Flatten
%Y A292421 Cf. A074712.
%K A292421 nonn,tabl
%O A292421 1,2
%A A292421 _Luc Rousseau_, Sep 16 2017