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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 (list; graph; refs; listen; history; text; internal format)
 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 Table of n, a(n) for n=2..25. 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: . * * * / \ / \ / \ *---* *---* *---* / \ / \ / \ * * * * * * / \ / \ / \ / \ / \ /#\ *---*---*---* *---*---*---* *---*---*---* /#####\ /#####\ /#####\#/#####\ /#####\ /#####\ *#######*#######* *#######*#######* *#######*#######* /#\#####/#\#####/#\ /#\#####/ \#####/#\ /#\#####/#\#####/ \ *---*---*---*---*---* *---*---*---*---*---* *---*---*---*---*---* . * * / \ / \ *---* *---* / \ / \ * * * * /#\ /#\ / \ /#\ *---*---*---* *---*---*---* /#####\ /#####\ /#####\#/#####\ *#######*#######* *#######*#######* / \#####/#\#####/ \ /#\#####/ \#####/ \ *---*---*---*---*---* *---*---*---*---*---* 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 Adjacent sequences: A356886 A356887 A356888 * A356890 A356891 A356892 KEYWORD nonn,easy AUTHOR Jack Hanke, Sep 02 2022 STATUS approved

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Last modified September 17 13:29 EDT 2024. Contains 375987 sequences. (Running on oeis4.)