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Number of length 6 nonnegative integer arrays starting and ending with 0 with adjacent elements differing by no more than n.
1

%I #8 Jun 06 2018 11:35:31

%S 21,120,404,1025,2181,4116,7120,11529,17725,26136,37236,51545,69629,

%T 92100,119616,152881,192645,239704,294900,359121,433301,518420,615504,

%U 725625,849901,989496,1145620,1319529,1512525,1725956,1961216,2219745

%N Number of length 6 nonnegative integer arrays starting and ending with 0 with adjacent elements differing by no more than n.

%C Row 5 of A204213.

%H R. H. Hardin, <a href="/A204214/b204214.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = (23/12)*n^4 + (37/6)*n^3 + (91/12)*n^2 + (13/3)*n + 1.

%F Conjectures from _Colin Barker_, Jun 06 2018: (Start)

%F G.f.: x*(21 + 15*x + 14*x^2 - 5*x^3 + x^4) / (1 - x)^5.

%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5.

%F (End)

%e Some solutions for n=5:

%e ..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0

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

%e ..1....4....2....2....3....2....2....6....1....5....8...10....8....2....3....5

%e ..4....3....7....1....2....0....3....5....5....4....6...10....3....6....3....6

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

%e ..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0

%Y Cf. A204213.

%K nonn

%O 1,1

%A _R. H. Hardin_, Jan 12 2012