Institution: Stanford University, Department of Computer Science

Title: A procedure for improving the upper bound for the number of n-ominoes.

Author: Klarner, David A.

Author: Rivest, Ronald L.

Date: February 1972

Abstract: An n-omino is a plane figure composed of n unit squares joined together along their edges. Every n-omino is generated by joining the edge of a unit square to the edge of a unit square in some (n-1)-omino so that the new square does not overlap any squares. Let t(n) denote the number of n-ominoes, then it is known that the sequence ${((t(n))}^{1/n} : n = 1,2,\ldots )$ increases to a limit $\Theta$ , and 3.72 < $\Theta$ < 6.75 . A procedure exists for computing an increasing sequence of numbers bounded above by $\Theta$. (Chandra recently showed that the limit of this sequence is $\Theta$ .) In the present work we give a procedure for computing a sequence of numbers bounded below by $\Theta$ . Whether or not the limit of this sequence is $\Theta$ remains an open question. By computing the first ten terms of our sequence, we have shown that $\Theta$ < 4.65 .

http://i.stanford.edu/pub/cstr/reports/cs/tr/72/263/CS-TR-72-263.pdf