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Number of length 1+3 0..n arrays with no four consecutive terms having the sum of any three elements equal to three times the fourth.
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%I #10 Nov 09 2018 21:55:00

%S 14,66,204,524,1098,2070,3584,5808,8934,13202,18828,26100,35306,46758,

%T 60792,77792,98118,122202,150476,183396,221442,265142,315000,371592,

%U 435494,507306,587652,677204,776610,886590,1007864,1141176,1287294

%N Number of length 1+3 0..n arrays with no four consecutive terms having the sum of any three elements equal to three times the fourth.

%H R. H. Hardin, <a href="/A249291/b249291.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-5) + 2*a(n-6) - 3*a(n-7) +a(n- 8).

%F Empirical for n mod 6 = 0: a(n) = n^4 + (8/3)*n^3 + 5*n^2 + 3*n

%F Empirical for n mod 6 = 1: a(n) = n^4 + (8/3)*n^3 + 5*n^2 + 3*n + (7/3)

%F Empirical for n mod 6 = 2: a(n) = n^4 + (8/3)*n^3 + 5*n^2 + 3*n + (8/3)

%F Empirical for n mod 6 = 3: a(n) = n^4 + (8/3)*n^3 + 5*n^2 + 3*n - 3

%F Empirical for n mod 6 = 4: a(n) = n^4 + (8/3)*n^3 + 5*n^2 + 3*n + (16/3)

%F Empirical for n mod 6 = 5: a(n) = n^4 + (8/3)*n^3 + 5*n^2 + 3*n - (1/3).

%F Empirical g.f.: 2*x*(7 + 12*x + 17*x^2 + 29*x^3 + 7*x^5) / ((1 - x)^5*(1 + x)*(1 + x + x^2)). - _Colin Barker_, Nov 09 2018

%e Some solutions for n=10:

%e 3 3 2 5 3 1 7 2 9 5 9 1 9 6 0 0

%e 3 7 0 9 5 7 10 0 8 9 2 5 5 7 1 5

%e 1 3 8 10 6 7 3 1 7 8 1 9 3 0 2 9

%e 1 1 4 6 0 3 10 9 0 2 10 8 4 4 10 1

%Y Row 1 of A249290.

%K nonn

%O 1,1

%A _R. H. Hardin_, Oct 24 2014