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A356889
a(n) = (n^2 + 3*n + 10/3)*4^(n-3) - 1/3.
0
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
2,1
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.
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
Cf. A334551.
Sequence in context: A144884 A004658 A034552 * A220616 A273803 A036754
KEYWORD
nonn,easy
AUTHOR
Jack Hanke, Sep 02 2022
STATUS
approved