OFFSET
2,1
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.
LINKS
Index entries for linear recurrences with constant coefficients, signature (13,-60,112,-64).
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)
EXAMPLE
a(3) = 21. Up to rotations and reflections, there are 5 possibilities:
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MATHEMATICA
Table[(n^2 + 3*n + 10/3)*4^(n-3) - 1/3, {n, 2, 25}] (* James C. McMahon, Jan 03 2024 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Jack Hanke, Sep 02 2022
STATUS
approved