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Number of 0..n arrays of length 7 with each element differing from at least one neighbor by 1 or less.
1

%I #7 Aug 09 2018 08:56:37

%S 128,1113,4240,11011,22912,41401,67936,103975,150976,210397,283696,

%T 372331,477760,601441,744832,909391,1096576,1307845,1544656,1808467,

%U 2100736,2422921,2776480,3162871,3583552,4039981,4533616,5065915,5638336

%N Number of 0..n arrays of length 7 with each element differing from at least one neighbor by 1 or less.

%C Row 7 of A221596.

%H R. H. Hardin, <a href="/A221599/b221599.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = 243*n^3 - 351*n^2 + 237*n + 127 for n>2.

%F Conjectures from _Colin Barker_, Aug 09 2018: (Start)

%F G.f.: x*(128 + 601*x + 556*x^2 + 217*x^3 - 16*x^4 - 28*x^5) / (1 - x)^4.

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>6.

%F (End)

%e Some solutions for n=6:

%e ..4....5....5....5....5....2....0....0....1....6....4....3....1....4....5....1

%e ..3....4....6....5....6....2....0....1....1....6....3....2....0....5....4....2

%e ..2....4....5....2....5....1....0....1....4....6....0....1....5....3....3....4

%e ..2....1....3....1....4....5....2....4....5....1....1....1....4....2....4....4

%e ..5....2....2....4....5....4....1....4....3....2....6....0....3....4....2....5

%e ..5....5....1....4....2....4....3....6....3....1....5....3....2....4....3....3

%e ..6....5....1....3....1....5....2....5....4....1....5....4....2....3....4....4

%Y Cf. A221596.

%K nonn

%O 1,1

%A _R. H. Hardin_, Jan 20 2013