A356888 a(n) = ((n-1)^2 + 2)*2^(n-2).

1, 3, 12, 44, 144, 432, 1216, 3264, 8448, 21248, 52224, 125952, 299008, 700416, 1622016, 3719168, 8454144, 19070976, 42729472, 95158272, 210763776, 464519168, 1019215872, 2227175424, 4848615424, 10519314432, 22749904896, 49056579584, 105495134208, 226291089408

Offset

1,2

Comments

a(n) is the number of fixed polyiamonds of minimal area 2*n-1 that touch each side of a triangle formed in the triangular lattice. n designates the number of triangles that touch each side of the larger triangle.

Links

Paolo Xausa, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Formula

G.f.: -x*(6*x^2-3*x+1)/(2*x-1)^3.

E.g.f.: (exp(2*x)*(3 - 2*x + 4*x^2) - 3)/4. - Stefano Spezia, Sep 02 2022

Example

a(3) = 12. Up to rotations and reflections there are 3 possibilities.
           *                     *                     *
          / \                   / \                   / \
         /   \                 /   \                 /   \
        *-----*               *-----*               *-----*
       / \   / \             / \   /#\             /#\   /#\
      /   \ /   \           /   \ /###\           /###\ /###\
     *-----*-----*         *-----*-----*         *-----*-----*
    /#\###/#\###/#\       /#\###/#\###/ \       / \###/#\###/ \
   /###\#/###\#/###\     /###\#/###\#/   \     /   \#/###\#/   \
  *-----*-----*-----*   *-----*-----*-----*   *-----*-----*-----*

Mathematica

A356888[n_] := ((n-1)^2 + 2)*2^(n-2); Array[A356888, 30] (* or *)
LinearRecurrence[{6, -12, 8}, {1, 3, 12}, 30] (* Paolo Xausa, Oct 07 2024 *)

Crossrefs

Cf. A334551.

Sequence in context: A290918 A012873 A282082 * A167477 A190051 A220633

Adjacent sequences: A356885 A356886 A356887 * A356889 A356890 A356891

Keyword

nonn,easy

Author

Jack Hanke, Sep 02 2022

Status

approved