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T(n,k) = Number of n X k integer arrays with each element equal to the number of horizontal, vertical, diagonal and antidiagonal neighbors less than itself.
5

%I #6 Jun 22 2021 14:35:58

%S 1,3,3,5,75,5,11,235,235,11,26,4267,2031,4267,26,55,72764,31940,31940,

%T 72764,55,119,527311,1002328,8205996,1002328,527311,119,263,7155883,

%U 17873307,661278762,661278762,17873307,7155883,263,573,100785131

%N T(n,k) = Number of n X k integer arrays with each element equal to the number of horizontal, vertical, diagonal and antidiagonal neighbors less than itself.

%C Table starts

%C ....1..........3..........5............11..............26..............55

%C ....3.........75........235..........4267...........72764..........527311

%C ....5........235.......2031.........31940.........1002328........17873307

%C ...11.......4267......31940.......8205996.......661278762.....50708665808

%C ...26......72764....1002328.....661278762...1028618143516.167704908247475

%C ...55.....527311...17873307...50708665808.167704908247475

%C ..119....7155883..329266534.8616991831997

%C ..263..100785131.7935241725

%C ..573.1027584101

%C .1248

%H R. H. Hardin, <a href="/A265960/b265960.txt">Table of n, a(n) for n = 1..60</a>

%F Empirical for column k:

%F k=1: a(n) = a(n-1) +a(n-2) +3*a(n-3) +a(n-4).

%F k=2: [order 30].

%e Some solutions for n=3, k=4

%e ..1..4..5..2....1..3..4..0....0..4..0..0....2..1..0..0....1..2..3..0

%e ..3..0..0..0....0..8..1..2....0..0..8..4....1..7..8..0....0..7..8..0

%e ..1..4..5..0....2..1..2..3....0..0..0..1....0..3..0..0....2..1..3..0

%Y Column 1 is A196423.

%K nonn,tabl

%O 1,2

%A _R. H. Hardin_, Dec 19 2015