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.
2, 3, 8, 4, 15, 64, 5, 22, 155, 1024, 6, 29, 246, 3151, 32768, 7, 36, 337, 5428, 127785, 2097152, 8, 43, 428, 7705, 237818, 10322065, 268435456, 9, 50, 519, 9982, 348849, 20729610, 1663418313, 68719476736, 10, 57, 610, 12259, 459880, 31374671
Offset: 1
Examples
Some solutions for n=4 k=4 ..3........2........0........2........3........0........0........3 ..2.3......2.3......0.0......1.1......3.3......0.0......1.0......2.3 ..3.2.2....2.2.2....1.0.0....2.1.1....2.3.3....0.1.0....0.0.1....3.3.4 ..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
Links
- R. H. Hardin, Table of n, a(n) for n = 1..205
Formula
Empirical for rows:
T(1,k) = 1*k + 1
T(2,k) = 7*k + 1
T(3,k) = 91*k - 27
T(4,k) = 2277*k - 1403 for k>1
T(5,k) = 111031*k - 95275 for k>2
T(6,k) = 10654607*k - 11243757 for k>3
T(7,k) = 2021888119*k - 2469384741 for k>4
Generalizing, T(n,k) = A195213(n) + const(n) for k>n-3
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
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