A316585 Number of ways to stack n triangles symmetrically (pointing upwards or downwards depending on row parity).

1, 1, 1, 2, 2, 3, 3, 6, 7, 12, 12, 21, 23, 39, 43, 74, 81, 138, 151, 257, 281, 479, 525, 895, 981, 1671, 1830, 3116, 3414, 5813, 6370, 10847, 11887, 20239, 22177, 37758, 41375, 70442, 77193, 131425, 144020, 245197, 268693, 457451, 501288, 853446, 935235, 1592242, 1744834, 2970580, 3255261

Offset

0,4

Links

Table of n, a(n) for n=0..50.

Example

a(9) = 12.
    *   *   *   *   *   *   *   *   *
   / \ / \ / \ / \ / \ / \ / \ / \ / \
  *---*---*---*---*---*---*---*---*---*
.
    *   *   *   *---*   *   *   *
   / \ / \ / \ / \ / \ / \ / \ / \
  *---*---*---*---*---*---*---*---*
.
    *   *   *---*---*   *   *
   / \ / \ / \ / \ / \ / \ / \
  *---*---*---*---*---*---*---*
.
    *   *---*   *   *---*   *
   / \ / \ / \ / \ / \ / \ / \
  *---*---*---*---*---*---*---*
.
    *---*   *   *   *   *---*
   / \ / \ / \ / \ / \ / \ / \
  *---*---*---*---*---*---*---*
.
    *   *---*---*---*   *
   / \ / \ / \ / \ / \ / \
  *---*---*---*---*---*---*
.
    *---*   *---*   *---*
   / \ / \ / \ / \ / \ / \
  *---*---*---*---*---*---*
.
    *---*---*---*---*
   / \ / \ / \ / \ / \
  *---*---*---*---*---*
.
          *   *
         / \ / \
    *   *---*---*   *
   / \ / \ / \ / \ / \
  *---*---*---*---*---*
.
      *           *
     / \         / \
    *---*   *   *---*
   / \ / \ / \ / \ / \
  *---*---*---*---*---*
.
      *       *
     / \     / \
    *---*---*---*
   / \ / \ / \ / \
  *---*---*---*---*
.
        *
       / \
      *---*
     / \ / \
    *---*---*
   / \ / \ / \
  *---*---*---*
.

Maple

Motzk := proc(x, y, twoar)
    option remember;
    if x =0 then
        if y <> 0 or twoar <>0 then
            return 0;
        else
            return 1;
        end if;
    elif y < 0 or y > x or twoar <x then
        return 0 ;
    elif y = 0 then
        if modp(x, 2) = 0 then
            return procname(x-1, y+1, twoar-2*y-1) ;
        else
            return 0 ;
        end if;
    elif modp(y, 2) = modp(x, 2) then
        return procname(x-1, y+1, twoar-2*y-1)
            +procname(x-1, y, twoar-2*y)
            +procname(x-1, y-1, twoar-2*y+1) ;
    else
        return procname(x-1, y, twoar-2*y) ;
    end if ;
end proc:
A316585 := proc(twoar)
    local a, x, y ;
    a:= 0 ;
    for x from 0 to twoar do
    for y from 0 to x do
        a := a+Motzk(x, y, twoar) ;
    end do:
    end do:
    a ;
end proc:
seq(A316585(n), n=0..50) ; # R. J. Mathar, Aug 23 2018

Mathematica

Motzk[x_, y_, twoar_] := Motzk[x, y, twoar] = Which[
x == 0, If[y != 0 || twoar != 0, 0, 1],
y < 0 || y > x || twoar < x, 0,
y == 0 , If[Mod[x, 2] == 0, Motzk[x - 1, y + 1, twoar - 2y - 1], 0],
Mod[y, 2] == Mod[x, 2], Motzk[x - 1, y + 1, twoar - 2y - 1] + Motzk[x - 1, y, twoar - 2y] + Motzk[x - 1, y - 1, twoar - 2y + 1],
True, Motzk[x - 1, y, twoar - 2y]];
A316585[twoar_] := Module[{a, x, y}, a = 0; For[x = 0, x <= twoar , x++, For[y = 0, y <= x, y++, a = a + Motzk[x, y, twoar]]]; a];
Table[A316585[n], {n, 0, 50}] (* Jean-François Alcover, Nov 08 2023, after R. J. Mathar *)

Crossrefs

Cf. A224704, A316384.

Sequence in context: A054172 A236971 A121211 * A369765 A146157 A238363

Adjacent sequences: A316582 A316583 A316584 * A316586 A316587 A316588

Keyword

nonn

Author

Seiichi Manyama, Jul 07 2018

Extensions

a(36)-a(50) from R. J. Mathar, Aug 23 2018

Status

approved