Jack Hanke - cp's OEIS frontend

User: Jack Hanke

Jack Hanke has authored 2 sequences.

A356889 a(n) = (n^2 + 3n + 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

Jack Hanke, Sep 02 2022

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.

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

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

Cf. A334551.

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

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)

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

Original entry on oeis.org

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

Jack Hanke, Sep 02 2022

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.

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

Paolo Xausa, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Cf. A334551.

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 *)

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

cp's OEIS Frontend

User: Jack Hanke

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

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

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cp's OEIS Frontend

User: Jack Hanke

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

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Mathematica

Formula

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

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Author

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Programs

Mathematica

Formula

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