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Number of length 1+5 0..n arrays with no disjoint triples in any consecutive six terms having the same sum.
1

%I #12 Nov 08 2018 06:33:57

%S 32,396,2292,9080,28020,72972,167576,349392,674520,1223180,2105772,

%T 3469896,5507852,8465100,12649200,18439712,26298576,36781452,50549540,

%U 68382360,91191012,120032396,156123912,200859120,255823880,322813452

%N Number of length 1+5 0..n arrays with no disjoint triples in any consecutive six terms having the same sum.

%H R. H. Hardin, <a href="/A247996/b247996.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = 6*a(n-1) - 14*a(n-2) + 14*a(n-3) - 14*a(n-5) + 14*a(n-6) - 6*a(n-7) + a(n-8).

%F Empirical for n mod 2 = 0: a(n) = n^6 + (1/2)*n^5 + (35/2)*n^4 - (5/2)*n^3 + 5*n^2 + 18*n

%F Empirical for n mod 2 = 1: a(n) = n^6 + (1/2)*n^5 + (35/2)*n^4 - (5/2)*n^3 + 5*n^2 + 18*n - (15/2).

%F Conjectures from _Colin Barker_, Nov 07 2018: (Start)

%F G.f.: 4*x*(8 + 51*x + 91*x^2 + 106*x^3 + 21*x^4 + 83*x^5) / ((1 - x)^7*(1 + x)).

%F a(n) = (2*n^6 + n^5 + 35*n^4 - 5*n^3 + 10*n^2 + 36*n) / 2 for n even.

%F a(n) = (2*n^6 + n^5 + 35*n^4 - 5*n^3 + 10*n^2 + 36*n - 15) / 2 for n odd.

%F (End)

%e Some solutions for n=6:

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

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

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

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

%e 3 2 2 5 2 2 2 1 1 0 5 5 3 5 3 2

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

%Y Row 1 of A247995.

%K nonn

%O 1,1

%A _R. H. Hardin_, Sep 28 2014