Comments on A026674, A026725, A026670
From: Rob Arthan
Date: Thu, 16 Jan 2003 14:31:13 +0000
A026674 can be obtained from either of the
2-dimensional arrays (A026670 and A026725). These
deal with very similar counting problems.
A026670 counts the paths from the origin to a given point in the
following diagram extending infinitely to the East and the South:
*+++++++ ....
+*++++++ ....
++*+++++ ....
+++*++++ ....
++++*+++ ....
+++++*++ ....
++++++*+ ....
+++++++* ....
++++++++ ....
.... ....
Here a path must go one step East or South but not both at a "+" and may go
one step East or South or both at a "*". (This is Manhattan taxicab geometry
including Broadway, i.e. a street that cuts diagonally across the
blocks).
A026725 counts the paths from the origin to a given point in a
similar diagram but with the diagonal street moved down one block (or the
origin moved up one block from Times Square!):
++++++++
*+++++++ ....
+*++++++ ....
++*+++++ ....
+++*++++ ....
++++*+++ ....
+++++*++ ....
++++++*+ ....
+++++++* ....
.... ....
A26674 deals with the case where the destination is of the form
(n+1, n) (taking (i, j) to mean i-th row and j-th column).