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%I #8 Apr 18 2022 10:27:36
%S 2,3,8,4,27,64,5,46,729,1024,6,65,1682,59049,32768,7,84,2729,190514,
%T 14348907,2097152,8,103,3776,357847,67379894,10460353203,268435456,9,
%U 122,4823,533142,147824001,74236765958,22876792454961,68719476736,10,141
%N T(n,k) = 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 two or less.
%C Table starts
%C .......2...........3...........4............5............6............7
%C .......8..........27..........46...........65...........84..........103
%C ......64.........729........1682.........2729.........3776.........4823
%C ....1024.......59049......190514.......357847.......533142.......709613
%C ...32768....14348907....67379894....147824001....237368212....329060365
%C .2097152.10460353203.74236765958.192172956591.333437946202.481573562101
%H R. H. Hardin, <a href="/A195248/b195248.txt">Table of n, a(n) for n = 1..132</a>
%F Empirical for rows:
%F T(1,k) = 1*k + 1,
%F T(2,k) = 19*k - 11
%F T(3,k) = 1047*k - 1459 for k>2,
%F T(4,k) = 176471*k - 349213 for k>4,
%F T(5,k) = 92031109*k - 223153377 for k>6,
%F T(6,k) = 149824887097*k - 417651128341 for k>8,
%F T(7,k) = 764465228592699*k - 2364216638005277 for k>10,
%F Generalizing, T(n,k) = A195214(n)*k + const(n) for k>2*n-4.
%e Some solutions for n=6, k=5
%e ..4............1............5............0............0............1
%e ..5.3..........1.3..........5.4..........1.2..........0.2..........0.0
%e ..5.4.5........1.2.4........3.4.4........1.3.1........1.0.0........2.1.2
%e ..4.3.4.4......3.3.3.3......5.3.5.4......2.1.1.2......0.1.1.0......0.0.2.2
%e ..5.4.2.2.4....4.5.4.5.5....4.3.5.4.3....2.3.1.3.1....2.0.2.2.0....2.0.1.2.4
%e ..4.3.2.2.4.4..3.5.3.3.5.5..3.5.5.3.3.1..4.2.2.1.1.0..0.2.0.2.0.1..1.1.0.2.4.4
%Y Column 1 is A006125(n+1).
%Y Column 2 is A047656(n+1).
%Y Cf. A195214.
%K nonn,tabl
%O 1,1
%A _R. H. Hardin_, Sep 13 2011
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