Illustration of a(13) by Hugo Pfoertner, using the comment in A289944:

There is a tiling with 13 equilateral triangles and John W. Layman's
calculated area of 1559 that has 13 triangles of sizes
2, 2, 3, 5, 7, 7, 7, 12, 14, 15, 15, 16, 18. - Peter Munn, Aug 24 2017

a(13) = 1559

                                *---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---*
                               / \                                                                     / \
                              +   +                                                                   +   +
                             /     \                                                                 /     \
                            +       +                                                               +       +
                           /         \                                                             /         \
                          +           +                                                           +           +
                         /             \                                                         /             \
                        +               +                                                       +               +
                       /                 \                                                     /                 \
                      +                   +                                                   +                   +
                     /                     \                                                 /                     \
                    +                       +                      18                       +                       +
                   /                         \                                             /                         \
                  +                           +                                           +                           +
                 /                             \                                         /                             \
                +                               +                                       +                               +
               /                                 \                                     /                                 \
              +                                   +                                   +                                   +
             /                                     \                                 /                                     \
            +                                       +                               +                                       +
           /                    15                   \                             /                                         \
          +                                           +                           +                                           +
         /                                             \                         /                                             \
        +                                               +                       +                      16                       +
       /                                                 \                     /                                                 \
      +                                                   +                   +                                                   +
     /                                                     \                 /                                                     \
    +                                                       +               +                                                       +
   /                                                         \             /                                                         \
  *---+---+---+---+---+---+---+---+---+---+---+---+---+---+---*           +                                                           +
   \                                                         / \         /                                                             \
    +                                                       +   +       *---+---*---+---+---+---+---+---+---+---+---+---+---+---+---+---*
     \                                                     /     \     / \  2  / \                                                     /
      +                                                   +   3   +   +   +   +   +                                                   +
       \                                                 /         \ /  2  \ /     \                                                 /
        +                                               *---+---+---*---+---*       +                                               +
         \                                             / \                 /         \                                             /
          +                     15                    +   +               +           +                                           +
           \                                         /     \             /             \                                         /
            +                                       +       +     5     +               +                 14                    +
             \                                     /         \         /                 \                                     /
              +                                   +           +       +         7         +                                   +
               \                                 /             \     /                     \                                 /
                +                               +               +   +                       +                               +
                 \                             /                 \ /                         \                             /
                  +                           +                   *---+---+---+---+---+---+---*                           +
                   \                         /                     \                         / \                         /
                    +                       +                       +                       +   +                       +
                     \                     /                         \                     /     \                     /
                      +                   +                           +                   +       +                   +
                       \                 /                             \        7        /         \                 /
                        +               +                               +               +           +               +
                         \             /                                 \             /             \             /
                          +           +                 12                +           +               +           +
                           \         /                                     \         /                 \         /
                            +       +                                       +       +         7         +       +
                             \     /                                         \     /                     \     /
                              +   +                                           +   +                       +   +
                               \ /                                             \ /                         \ /
                                *---+---+---+---+---+---+---+---+---+---+---+---*---+---+---+---+---+---+---*