A091990 - OEIS

%I #1 Jun 12 2004 03:00:00

%S 1,2,10,140,185,28490,3568,4117006,68166,695025466,1324149

%N Number of reduced magic borders of order n (these extend magic squares of order n-2 to order n).

%C Definition for a magic border: imagine the border of a square matrix: E_1={a_{1,1},...,a_{1,n}} upper, E_2={a_{1,1},...,a_{n,1}} left, E_3={a_{n,1},...,a_{n,n}} lower, E_4={a_{1,n},...,a_{n,n}} right edge. Distribute the pairs of numbers (0,n^2-1),...,(2n-3,n^2-2n+2) (each once) on (a_{1,1},a_{n,n}), (a_{1,n},a_{n,1}) the pairs of corner elements and (a_{1,i},a_{n,i}), (a_{i,1},a_{i,n}) (i=2,...,n-1) the pairs of edges so that for k=1,...,4: sum_{a\in E_k) a = const. Reduced means that the pairs of corners, (a_{1,2},...,a_{1,n-1}) and (a_{2,1},...,a_{n-1,1}) are in ascending order.

%D M. Kraitchik, Bordered Squares. Section 7.7 in Mathematical Recreations, Dover, NY, 2nd ed., 1953, pp. 167-170

%H E. S. de Cabezon, <a href="http://arXiv.org/abs/math.CO/0305271">Bordered magic squares: elements for a comprehensive approach</a>.

%K more,nonn

%O 3,2

%A Andreas W. Reinhart (reinhart(AT)castor.uni-trier.de), Mar 17 2004