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A194998 T(n,k)=Number of lower triangles of an n X n 0..k array with each element differing from all of its horizontal and vertical neighbors by one.

%I A194998 #8 Jul 22 2025 12:39:57
%S A194998 2,3,2,4,6,2,5,10,20,2,6,14,42,80,2,7,18,66,248,576,2,8,22,90,458,
%T A194998 2290,4608,2,9,26,114,672,4990,31042,69632,2,10,30,138,888,7858,81014,
%U A194998 641376,1114112,2,11,34,162,1104,10804,138956,2059822,19753266,34603008,2,12,38
%N A194998 T(n,k)=Number of lower triangles of an n X n 0..k array with each element differing from all of its horizontal and vertical neighbors by one.
%C A194998 Table starts
%C A194998 .2.......3........4........5.........6.........7.........8.........9........10
%C A194998 .2.......6.......10.......14........18........22........26........30........34
%C A194998 .2......20.......42.......66........90.......114.......138.......162.......186
%C A194998 .2......80......248......458.......672.......888......1104......1320......1536
%C A194998 .2.....576.....2290.....4990......7858.....10804.....13754.....16706.....19658
%C A194998 .2....4608....31042....81014....138956....199988....261324....322806....384292
%C A194998 .2...69632...641376..2059822...3816148...5740166...7686580...9643120..11600228
%C A194998 .2.1114112.19753266.78060014.159427052.251606900.346018484.441301618.536657628
%H A194998 R. H. Hardin, <a href="/A194998/b194998.txt">Table of n, a(n) for n = 1..373</a>
%F A194998 Empirical for rows:
%F A194998 T(1,k) = 1*k + 1
%F A194998 T(2,k) = 4*k - 2
%F A194998 T(3,k) = 24*k - 30 for k>2
%F A194998 T(4,k) = 216*k - 408 for k>4
%F A194998 T(5,k) = 2952*k - 6910 for k>6
%F A194998 T(6,k) = 61488*k - 169100 for k>8
%F A194998 T(7,k) = 1957392*k - 6016308 for k>10
%F A194998 Generalizing, T(n,k) = A145237(n)*k + const(n), for k>2*n-4
%e A194998 Some solutions for n=4 k=4
%e A194998 ..3........2........3........0........1........4........4........3
%e A194998 ..2.3......1.0......2.1......1.0......2.1......3.2......3.2......2.1
%e A194998 ..1.2.1....0.1.2....3.2.3....2.1.2....1.0.1....2.1.2....4.3.2....1.2.3
%e A194998 ..0.1.0.1..1.2.1.0..2.3.2.3..1.0.1.2..2.1.2.1..1.2.1.0..3.2.3.4..2.1.2.1
%K A194998 nonn,tabl
%O A194998 1,1
%A A194998 _R. H. Hardin_ Sep 07 2011