[seqfan] Guess the formula T(n,k)=?

[Ron Hardin]

This is likely to be T(n,k)=f(n)+f(k) with f() having a 2^() flavor.


Every row, column and the diagonal satisfy the same recurrence.

The solutions seem to pick every 3rd row or column for 1's and sometimes add cross rows or columns of 1's.


/tmp/eyg
T(n,k)=Number of (n+2)X(k+2) 0..1 arrays with no 3x3 subblock diagonal sum 0 and no antidiagonal sum 0 and no row sum 2 and no column sum 2

Table starts
..23..22..31..43..61..88.127.184.268.391.571..835.1222.1789.2620.3838.5623.8239
..22..28..37..49..67..94.133.190.274.397.577..841.1228.1795.2626.3844.5629.8245
..31..37..46..58..76.103.142.199.283.406.586..850.1237.1804.2635.3853.5638.8254
..43..49..58..70..88.115.154.211.295.418.598..862.1249.1816.2647.3865.5650.8266
..61..67..76..88.106.133.172.229.313.436.616..880.1267.1834.2665.3883.5668.8284
..88..94.103.115.133.160.199.256.340.463.643..907.1294.1861.2692.3910.5695.8311
.127.133.142.154.172.199.238.295.379.502.682..946.1333.1900.2731.3949.5734.8350
.184.190.199.211.229.256.295.352.436.559.739.1003.1390.1957.2788.4006.5791.8407
.268.274.283.295.313.340.379.436.520.643.823.1087.1474.2041.2872.4090.5875.8491
.391.397.406.418.436.463.502.559.643.766.946.1210.1597.2164.2995.4213.5998.8614

Empirical for diagonal:
a(n)=2*a(n-1)-a(n-2)+a(n-3)-a(n-4) for n>5
Empirical for column k:
k=1: a(n)=2*a(n-1)-a(n-2)+a(n-3)-a(n-4) for n>5
k=2: a(n)=2*a(n-1)-a(n-2)+a(n-3)-a(n-4)
k=3: a(n)=2*a(n-1)-a(n-2)+a(n-3)-a(n-4)
k=4: a(n)=2*a(n-1)-a(n-2)+a(n-3)-a(n-4)
k=5: a(n)=2*a(n-1)-a(n-2)+a(n-3)-a(n-4)
k=6: a(n)=2*a(n-1)-a(n-2)+a(n-3)-a(n-4)
k=7: a(n)=2*a(n-1)-a(n-2)+a(n-3)-a(n-4)

Two solutions for n=8 k=8
..0..0..0..0..0..1..0..0..1..0....0..1..0..0..1..0..0..1..0..0
..1..1..1..1..1..1..1..1..1..1....0..1..0..0..1..0..0..1..0..0
..0..0..0..0..0..1..0..0..1..0....0..1..0..0..1..0..0..1..0..0
..0..0..0..0..0..1..0..0..1..0....1..1..1..1..1..1..1..1..1..1
..1..1..1..1..1..1..1..1..1..1....0..1..0..0..1..0..0..1..0..0
..0..0..0..0..0..1..0..0..1..0....0..1..0..0..1..0..0..1..0..0
..0..0..0..0..0..1..0..0..1..0....1..1..1..1..1..1..1..1..1..1
..1..1..1..1..1..1..1..1..1..1....0..1..0..0..1..0..0..1..0..0
..0..0..0..0..0..1..0..0..1..0....0..1..0..0..1..0..0..1..0..0
..0..0..0..0..0..1..0..0..1..0....0..1..0..0..1..0..0..1..0..0


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