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Number of length 1+4 0..n arrays with no pair in any consecutive five terms totalling exactly n.
1

%I #7 Nov 06 2018 11:50:04

%S 2,12,124,424,1566,3876,9368,18768,36250,63100,106452,168312,259574,

%T 383124,554416,777376,1072818,1445868,1923500,2512200,3245902,4132612,

%U 5214024,6499824,8040266,9846876,11979268,14450968,17331750,20637300

%N Number of length 1+4 0..n arrays with no pair in any consecutive five terms totalling exactly n.

%H R. H. Hardin, <a href="/A246738/b246738.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = 3*a(n-1) - 8*a(n-3) + 6*a(n-4) + 6*a(n-5) - 8*a(n-6) + 3*a(n-8) - a(n-9).

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

%F G.f.: 2*x*(1 + 3*x + 44*x^2 + 34*x^3 + 189*x^4 + 43*x^5 + 166*x^6) / ((1 - x)^6*(1 + x)^3).

%F a(n) = 10*n - 20*n^2 + 15*n^3 - 5*n^4 + n^5 for n even.

%F a(n) = 16 - 15*n - 10*n^2 + 15*n^3 - 5*n^4 + n^5 for n odd.

%F (End)

%e Some solutions for n=4:

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

%e ..0....0....0....3....1....3....1....4....3....4....3....3....3....3....4....3

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

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

%e ..1....0....0....3....0....0....1....1....3....3....4....3....3....4....1....4

%Y Row 1 of A246737.

%K nonn

%O 1,1

%A _R. H. Hardin_, Sep 02 2014