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T(n,k)=Number of nXnXn triangular 0..k arrays with some element plus some adjacent element totalling k exactly once.
12

%I #4 Mar 14 2016 09:17:24

%S 0,0,0,0,12,0,0,24,66,0,0,48,768,468,0,0,72,4428,37848,3612,0,0,108,

%T 14976,772056,3280968,40020,0,0,144,42750,7876728,308256072,534438768,

%U 601368,0,0,192,96768,51535116,12712991544,302595682944,168922341960

%N T(n,k)=Number of nXnXn triangular 0..k arrays with some element plus some adjacent element totalling k exactly once.

%C Table starts

%C .0.....0.........0............0..............0................0

%C .0....12........24...........48.............72..............108

%C .0....66.......768.........4428..........14976............42750

%C .0...468.....37848.......772056........7876728.........51535116

%C .0..3612...3280968....308256072....12712991544.....233617868244

%C .0.40020.534438768.302595682944.67475902622400.4283012188676772

%H R. H. Hardin, <a href="/A270256/b270256.txt">Table of n, a(n) for n = 1..98</a>

%F Empirical for row n:

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

%F n=3: [order 10]

%F n=4: [order 18]

%F Empirical quasipolynomials for row n:

%F n=2: polynomial of degree 2 plus a quasipolynomial of degree 0 with period 2

%F n=3: polynomial of degree 5 plus a quasipolynomial of degree 3 with period 2

%F n=4: polynomial of degree 9 plus a quasipolynomial of degree 7 with period 2

%e Some solutions for n=3 k=4

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

%e ...1.1....1.3....0.0....4.1....4.3....4.0....3.2....2.1....4.3....4.2....1.0

%e ..0.4.4..1.2.0..2.2.1..3.1.2..3.0.2..2.1.0..4.0.1..1.4.2..3.0.3..3.3.4..1.0.2

%K nonn,tabl

%O 1,5

%A _R. H. Hardin_, Mar 14 2016