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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.
2
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
1,5
COMMENTS
A polyomino with a width of m and height of n must have at least m + n - 1 cells.
LINKS
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
EXAMPLE
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 ...
...
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 *)
PROG
(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
CROSSREFS
Columns 2..3 are A008574(n-1), A164754(n+1).
Main diagonal is A334551.
Cf. A292357.
Sequence in context: A289511 A158687 A141541 * A347676 A177947 A132789
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Jun 06 2020
STATUS
approved