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A194540 T(n,k) is the number of lower triangles of an n X n 0..k array with new values introduced in row major order 0..k and no element equal to more than one horizontal or vertical neighbor.

%I A194540 #9 Oct 16 2023 14:53:03
%S A194540 1,1,3,1,4,6,1,4,59,12,1,4,116,1851,23,1,4,131,16556,119548,43,1,4,
%T A194540 132,43785,7721920,16039294,79,1,4,132,62038,79201795,11525456507,
%U A194540 4460214471,143,1,4,132,67377,286773762,627974590650,54979732214116,2572187445993
%N A194540 T(n,k) is the number of lower triangles of an n X n 0..k array with new values introduced in row major order 0..k and no element equal to more than one horizontal or vertical neighbor.
%H A194540 R. H. Hardin, <a href="/A194540/b194540.txt">Table of n, a(n) for n = 1..86</a>
%e A194540 Table starts:
%e A194540 ...1.............1..................1......................1
%e A194540 ...3.............4..................4......................4
%e A194540 ...6............59................116....................131
%e A194540 ..12..........1851..............16556..................43785
%e A194540 ..23........119548............7721920...............79201795
%e A194540 ..43......16039294........11525456507...........627974590650
%e A194540 ..79....4460214471.....54979732214116......21399815343394998
%e A194540 .143.2572187445993.838387866908847478.3132305932642809585983
%e A194540 Some solutions for n=4 and k=3:
%e A194540 ..0........0........0........0........0........0........0........0
%e A194540 ..1.2......1.0......1.0......1.0......0.1......1.2......1.0......1.2
%e A194540 ..3.1.1....0.1.0....2.0.3....2.3.1....2.3.0....1.3.3....2.3.3....0.3.1
%e A194540 ..2.3.2.0..2.1.3.1..2.1.0.1..1.0.3.2..3.0.3.1..3.1.1.2..0.2.2.0..2.1.2.1
%Y A194540 Column 1 is A055244.
%K A194540 nonn,tabl
%O A194540 1,3
%A A194540 _R. H. Hardin_, Aug 28 2011