# [seqfan] Prove formula? Simple formula, difficult enumeration

Ron Hardin rhhardin at att.net
Thu Mar 29 16:12:21 CEST 2012

```Can the empirical formula be proven?  The enumeration is hard to extend far
because of array size, but the emerging formula is simple.

T(n,k)=Number of (n+1)X(n+1) -k..k symmetric matrices with every 2X2 subblock
having sum zero

Empirical: T(n,k)=k^(n+1)+(k+1)^(n+1)

Table starts
....5....13....25....41....61.....85...113...145..181.221
....9....35....91...189...341....559...855..1241.1729....
...17....97...337...881..1921...3697..6497.10657.........
...33...275..1267..4149.10901..24583.49575...............
...65...793..4825.19721.62281.164305.....................
..129..2315.18571.94509..................................
..257..6817.72097........................................
..513.20195..............................................
.1025....................................................

Some solutions for n=3 k=4
.-2..1.-3..0....0.-1..0..1....4..0..1.-1....2.-1.-1.-2....3.-2..1..0..
..1..0..2..1...-1..2.-1..0....0.-4..3.-3...-1..0..2..1...-2..1..0.-1..
.-3..2.-4..1....0.-1..0..1....1..3.-2..2...-1..2.-4..1....1..0.-1..2..
..0..1..1..2....1..0..1.-2...-1.-3..2.-2...-2..1..1..2....0.-1..2.-3..

rhhardin at mindspring.com
rhhardin at att.net (either)

```