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A214540 T(n,k)=Number of nXnXn triangular 0..k arrays with no element lying outside the (possibly reversed) range delimited by its sw and se neighbors, and every horizontal row having the same average value.

%I A214540 #7 Jul 22 2025 23:17:27
%S A214540 2,3,2,4,5,2,5,8,15,2,6,13,38,93,2,7,18,79,344,1007,2,8,25,152,1181,
%T A214540 5360,17213,2,9,32,263,3198,35567,141470,461465,2,10,41,418,7801,
%U A214540 155102,2280331,6042900,19166997,2,11,50,643,16752,562559,15986168,292689331
%N A214540 T(n,k)=Number of nXnXn triangular 0..k arrays with no element lying outside the (possibly reversed) range delimited by its sw and se neighbors, and every horizontal row having the same average value.
%C A214540 Table starts
%C A214540 .2....3....4.....5......6......7.......8.......9.......10.....11...12.13.14
%C A214540 .2....5....8....13.....18.....25......32......41.......50.....61...72.85
%C A214540 .2...15...38....79....152....263.....418.....643......942...1329.1832
%C A214540 .2...93..344..1181...3198...7801...16752...33605....62766.111653
%C A214540 .2.1007.5360.35567.155102.562559.1786114.4984385.12779966
%H A214540 R. H. Hardin, <a href="/A214540/b214540.txt">Table of n, a(n) for n = 1..85</a>
%F A214540 Empirical for row n:
%F A214540 n=1: a(k)=2*a(k-1)-a(k-2)
%F A214540 n=2: a(k)=2*a(k-1)-2*a(k-3)+a(k-4)
%F A214540 n=3: a(k)=2*a(k-1)-3*a(k-4)+3*a(k-6)-2*a(k-9)+a(k-10)
%F A214540 n=4: (symmetric, order 30)
%e A214540 Some solutions for n=4 k=4
%e A214540 .....3........2........3........3........1........1........2........2
%e A214540 ....2.4......2.2......2.4......3.3......2.0......0.2......2.2......2.2
%e A214540 ...2.3.4....2.3.1....2.3.4....3.3.3....2.1.0....0.2.1....3.1.2....3.0.3
%e A214540 ..3.2.3.4..1.3.0.4..3.2.4.3..4.3.1.4..2.1.1.0..1.0.2.1..3.1.1.3..3.0.3.2
%Y A214540 Row 2 is A000982(n+1)
%K A214540 nonn,tabl
%O A214540 1,1
%A A214540 _R. H. Hardin_ Jul 20 2012