A356889 a(n) = (n^2 + 3*n + 10/3)*4^(n-3) - 1/3.

3, 21, 125, 693, 3669, 18773, 93525, 456021, 2184533, 10310997, 48059733, 221599061, 1012225365, 4585772373, 20624790869, 92162839893, 409453548885, 1809612887381, 7960006055253, 34863681197397, 152099108509013, 661172992169301, 2864594294232405, 12373170851239253

Offset: 2

Author

Jack Hanke, Sep 02 2022

Keywords

• nonn
• easy

Comments

a(n) is the number of fixed polyforms of minimal area (2*n)-1 that contain at least one triangle that touches each side of a triangle formed on a Kagome (trihexagonal) lattice. n is the number of triangles that touch each side of the larger triangle.

Examples

			a(3) = 21. Up to rotations and reflections, there are 5 possibilities:
.
            *                      *                      *
           / \                    / \                    / \
          *---*                  *---*                  *---*
         /     \                /     \                /     \
        *       *              *       *              *       *
       / \     / \            / \     / \            / \     /#\
      *---*---*---*          *---*---*---*          *---*---*---*
     /#####\ /#####\        /#####\#/#####\        /#####\ /#####\
    *#######*#######*      *#######*#######*      *#######*#######*
   /#\#####/#\#####/#\    /#\#####/ \#####/#\    /#\#####/#\#####/ \
  *---*---*---*---*---*  *---*---*---*---*---*  *---*---*---*---*---*
.
            *                      *
           / \                    / \
          *---*                  *---*
         /     \                /     \
        *       *              *       *
       /#\     /#\            / \     /#\
      *---*---*---*          *---*---*---*
     /#####\ /#####\        /#####\#/#####\
    *#######*#######*      *#######*#######*
   / \#####/#\#####/ \    /#\#####/ \#####/ \
  *---*---*---*---*---*  *---*---*---*---*---*
		

Links

• Index entries for linear recurrences with constant coefficients, signature (13,-60,112,-64).

Crossrefs

Cf. A334551.

Programs

Mathematica

Table[(n^2 + 3*n + 10/3)*4^(n-3) - 1/3, {n,2,25}] (* James C. McMahon, Jan 03 2024 *)

Formula

G.f.: x^2*(3 - 18*x + 32*x^2 - 8*x^3)/((1 - x)*(1 - 4*x)^3). - adapted to the offset by Stefano Spezia, Sep 03 2022

From Stefano Spezia, Sep 03 2022: (Start)

a(n) = (4^n*(10 + 3*n*(3 + n)) - 64)/192.

a(n) = 13*a(n-1) - 60*a(n-2) + 112*a(n-3) - 64*a(n-4) for n > 5. (End)