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A195232 T(n,k)is the number of lower triangles of an n X n 0..k array with each element differing from all of its diagonal, vertical, antidiagonal and horizontal neighbors by one or less.

%I A195232 #16 Jul 22 2025 12:41:48
%S A195232 2,3,8,4,15,64,5,22,155,1024,6,29,246,3151,32768,7,36,337,5428,127785,
%T A195232 2097152,8,43,428,7705,237818,10322065,268435456,9,50,519,9982,348849,
%U A195232 20729610,1663418313,68719476736,10,57,610,12259,459880,31374671
%N A195232 T(n,k)is the number of lower triangles of an n X n 0..k array with each element differing from all of its diagonal, vertical, antidiagonal and horizontal neighbors by one or less.
%C A195232 Table starts
%C A195232 ...........2............3.............4.............5.............6
%C A195232 ...........8...........15............22............29............36
%C A195232 ..........64..........155...........246...........337...........428
%C A195232 ........1024.........3151..........5428..........7705..........9982
%C A195232 .......32768.......127785........237818........348849........459880
%C A195232 .....2097152.....10322065......20729610......31374671......42029278
%C A195232 ...268435456...1663418313....3601738548....5618308863....7640055854
%C A195232 .68719476736.535153390177.1249159521262.2006626824777.2767861764930
%H A195232 R. H. Hardin, <a href="/A195232/b195232.txt">Table of n, a(n) for n = 1..205</a>
%F A195232 Empirical for rows:
%F A195232 T(1,k) = 1*k + 1
%F A195232 T(2,k) = 7*k + 1
%F A195232 T(3,k) = 91*k - 27
%F A195232 T(4,k) = 2277*k - 1403 for k>1
%F A195232 T(5,k) = 111031*k - 95275 for k>2
%F A195232 T(6,k) = 10654607*k - 11243757 for k>3
%F A195232 T(7,k) = 2021888119*k - 2469384741 for k>4
%F A195232 Generalizing, T(n,k) = A195213(n) + const(n) for k>n-3
%F A195232 Since elements of a solution differ by no more than n, T(n,k)-T(n,k-1) is constant for k >= n. This confirms the empirical formula: T(n,k) is a polynomial of degree 1 in k for k > n-3. - _Robert Israel_, Nov 21 2017
%e A195232 Some solutions for n=4 k=4
%e A195232 ..3........2........0........2........3........0........0........3
%e A195232 ..2.3......2.3......0.0......1.1......3.3......0.0......1.0......2.3
%e A195232 ..3.2.2....2.2.2....1.0.0....2.1.1....2.3.3....0.1.0....0.0.1....3.3.4
%e A195232 ..2.3.2.3..1.2.3.3..0.1.1.0..1.1.2.1..3.3.3.2..0.0.0.0..1.1.1.0..3.4.4.3
%K A195232 nonn,tabl
%O A195232 1,1
%A A195232 _R. H. Hardin_, Sep 13 2011