A334551 -id:A334551 - cp's OEIS frontend

A334552 Array read by antidiagonals: T(m,n) is the number of fixed polyominoes that have a width of m and height of n and m + n - 1 cells.

1, 1, 1, 1, 4, 1, 1, 8, 8, 1, 1, 12, 25, 12, 1, 1, 16, 50, 50, 16, 1, 1, 20, 83, 120, 83, 20, 1, 1, 24, 124, 230, 230, 124, 24, 1, 1, 28, 173, 388, 497, 388, 173, 28, 1, 1, 32, 230, 602, 932, 932, 602, 230, 32, 1, 1, 36, 295, 880, 1591, 1924, 1591, 880, 295, 36, 1

Offset: 1

Andrew Howroyd, Jun 06 2020

A polyomino with a width of m and height of n must have at least m + n - 1 cells.

			Array begins:
=====================================================
m\n | 1  2   3    4    5     6     7     8      9
----+------------------------------------------------
  1 | 1  1   1    1    1     1     1     1      1 ...
  2 | 1  4   8   12   16    20    24    28     32 ...
  3 | 1  8  25   50   83   124   173   230    295 ...
  4 | 1 12  50  120  230   388   602   880   1230 ...
  5 | 1 16  83  230  497   932  1591  2538   3845 ...
  6 | 1 20 124  388  932  1924  3588  6212  10156 ...
  7 | 1 24 173  602 1591  3588  7265 13582  23859 ...
  8 | 1 28 230  880 2538  6212 13582 27288  51290 ...
  9 | 1 32 295 1230 3845 10156 23859 51290 102745 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Columns 2..3 are A008574(n-1), A164754(n+1).
Main diagonal is A334551.
Cf. A292357.

A334552[m_,n_]:=Max[1,8Binomial[m+n-2,m-1]-3m*n+2m+2n-8];
Table[A334552[m-n+1,n],{m,15},{n,m}] (* Paolo Xausa, Dec 20 2023 *)

T(m, n)={if(m==1||n==1, 1, 8*binomial(m+n-2, m-1) - 3*m*n + 2*m + 2*n - 8)} \\ Andrew Howroyd, Dec 30 2020, after Peter J. Taylor

T(m,n) = 2*binomial(m+n-2, m-1) + 2*(m+n-4) + (m-2)*(n-2)*(m+n-5) + 2*Sum_{i=1..m-2} Sum_{j=1..n-2} ((m-2-i)*(n-2-j)+2)*binomial(i+j,i) for m > 1, n > 1.

T(m,n) = max(1, 8*binomial(m+n-2, m-1) - 3*m*n + 2*m + 2*n - 8). - Peter J. Taylor, Dec 15 2020

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

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

Original entry on oeis.org

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)

cp's OEIS Frontend

A334552 Array read by antidiagonals: T(m,n) is the number of fixed polyominoes that have a width of m and height of n and m + n - 1 cells.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

cp's OEIS Frontend

A334552 Array read by antidiagonals: T(m,n) is the number of fixed polyominoes that have a width of m and height of n and m + n - 1 cells.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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