OFFSET
0,3
COMMENTS
Computed using Burnside's orbit-counting lemma.
LINKS
María Merino, Rows n=0..46 of triangle, flattened
M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).
FORMULA
For even n and m: T(n,m) = (3^(m*n) + 3*3^(m*n/2))/4;
for even n and odd m: T(n,m) = (3^(m*n) + 3^((m*n+n)/2) + 2*3^(m*n/2))/4;
for odd n and even m: T(n,m) = (3^(m*n) + 3^((m*n+m)/2) + 2*3^(m*n/2))/4;
for odd n and m: T(n,m) = (3^(m*n) + 3^((m*n+n)/2) + 3^((m*n+m)/2) + 3^((m*n+1)/2))/4.
EXAMPLE
Triangle begins:
===========================================================
n\ m | 0 1 2 3 4 5
-----|-----------------------------------------------------
0 | 1
1 | 1 3
2 | 1 6 27
3 | 1 18 216 5346
4 | 1 45 1701 134865 10766601
5 | 1 135 15066 3608550 871858485 211829725395
...
MATHEMATICA
Table[Which[AllTrue[{n, m}, EvenQ], (3^(m n)+3 3^((m n)/2))/4, EvenQ[ n]&&OddQ[m], (3^(m n)+3^((m n+n)/2)+2 3^((m n)/2))/4, OddQ[n]&&EvenQ[ m], (3^(m n)+3^((m n+m)/2)+2 3^((m n)/2))/4, True, (3^(m n)+3^((m n+n)/2)+3^((m n+m)/2)+3^((m n+1)/2))/4], {n, 0, 10}, {m, 0, n}]//Flatten (* Harvey P. Dale, Mar 29 2023 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
María Merino, Imanol Unanue, Yosu Yurramendi, May 15 2017
STATUS
approved