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

Author

Andrew Howroyd, Jun 06 2020

Keywords

• nonn
• tabl

Comments

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

Examples

			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 ...
  ...
		

Links

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

Crossrefs

Columns 2..3 are A008574(n-1), A164754(n+1).

Main diagonal is A334551.

Cf. A292357.

Programs

Mathematica

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 *)

PARI

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

Formula

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