Hexa/tria lattice generation – Mathematica Stack Exchange

If I understood you question correctly, then the pattern consists of lines of 3 types:

  1. vertical lines through points x= 0,2,4,…, y=0
  2. slanted lines with angle of 30 degrees against the horizontal through points: x=1,5,9,13,.., y=0
    3)slanted lines with angle of -30 degrees against the horizontal through points: x=1,5,9,13,.., y=0

We may draw this by:

n = 15;
v = 0.5;(*slope*)
linesvert = Table(InfiniteLine({4 i, 0}, {1, v}), {i, n});
linesslant1 = Table(InfiniteLine({4 i, 0}, {-1, v}), {i, n});
linesslant2 = 
  Table({InfiniteLine({1 + 4 i, 0}, {0, 1}), 
    InfiniteLine({3 + 4 i, 0}, {0, 1})}, {i, n});
Graphics({
  linesvert, linesslant1, linesslant2
  , Red, Thick, Line({{21, v}, {24, 4 v}, {21, 7 v}, {21, v}}), 
  Line({{20, 4 v}, {23, 1 v}, {23, 7 v}, {20, 4 v}})
  }, PlotRange -> {{15, 40}, {0, 8}})

enter image description here