Counting Kings: Explicit Formulas, Recurrence Relations, and Generating Functions! Oh My! Neil J. Calkin, Kevin James, Shannon Purvis, Shaina Race, Clemson; Keith Schneider, UNCA; Matthew Yancey, Virginia Tech July 18, 2006 Abstract Let F (m, n) be the number of distinct configurations any num- ber of non-attacking kings on an m × n chessboard. We’ll explore the sequences generated when we fix n at some small value and dis- cuss methods of finding explicit formulas, recurrence relations and generating functions for these sequences. 1 Introduction Consider the function in two variables F (m, n)= 1 T A m−1 n 1 where A n is a recursively defined matrix as follows: A 0 = (1), A 1 =  1 1 1 0  , and A n =  A n−1 A n−2 A n−2 0  with the copies of A n−2 padded with zeros as needed and 0 being a zero matrix of the appropriate size. Additionally, 1 T and 1 are respectively a row and column vector of all ones of the necessary length for multiplicative conformity. In other words F (m, n) is the sum of the elements of A n raised to the m − 1 power. F (m, n) gives the number of distinct configurations of an arbitrary num- ber of non-attacking kings on an m × n chessboard. This was derived using techniques found in Calkin and Wilf[2] but proof of this will not be pre- sented herein. F (m, n) also counts the distinct tilings of an (m+1) × (n +1) rectangle with 1×1 and 2×2 tiles which is a problem studied by Heubach[3]. This research was funded, in part, by a grant from the NSF. 1