And... connections, connections, Neil Sloane and I work in the same building, recently named Shannon Lab! And here is a link linking Golomb, Sloane and Shannon!
The F Pentomino Is A MethuselahIn John Conway's Game of Life, the F pentomino has a secure reputation as a methuselah. If you don't know about Conway's Game of Life, it is well worth investigating. Many sites provide Java applets. In particular, Alan Hensel. There you can open RPENTO.LIF and play it. The GoL people keep on calling it the R pentomino... what do they know!
Counting Poly-ominoesThere is one domino and 12 pentominoes. In trying to count the N-ominoes, nobody has been able to go much beyond the 24-ominoes. This series of numbers is a documented Integer Series. To quote from another site :The number of n cell polyominoes (that allow holes) for n = 1,2,...,15, is 1, 1, 2, 5, 12, 35, 108, 369, 1285, 4655, 17073, 63600, 238951, 901971, 3426576.
This is sequence A000105(M1425) in Sloane's database of integer sequences.
The number of n cell polyominoes (without holes) for n = 1,2,...,15, is 1, 1, 2, 5, 12, 35, 107, 363, 1248, 4460, 16094, 58937, 217117, 805475, 3001211. This is sequence A000104(M1424) in Sloane's database of integer sequences.
The program used is due to John Boyer, and is based on a paper of Redelmeier (Discrete Math. 36 (1981) 191-203).
Now, from percolation theory to java... hmm, I don't smell a real connection there!