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 A216938 Number of side-2 hexagonal 0..n arrays with values nondecreasing E, SW and SE 1

%I #9 Jul 23 2018 08:44:33

%S 10,53,200,606,1572,3630,7656,15015,27742,48763,82160,133484,210120,

%T 321708,480624,702525,1006962,1418065,1965304,2684330,3617900,4816890,

%U 6341400,8261955,10660806,13633335,17289568,21755800,27176336,33715352

%N Number of side-2 hexagonal 0..n arrays with values nondecreasing E, SW and SE

%C Row 2 of A216937.

%H R. H. Hardin, <a href="/A216938/b216938.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = (1/1260)*n^7 + (1/60)*n^6 + (53/360)*n^5 + (17/24)*n^4 + (719/360)*n^3 + (131/40)*n^2 + (1199/420)*n + 1= (n+5) *(n+4) *(n+3) *(n+2) *(n+1) *(2*n^2+12*n+21)/2520.

%F Conjectures from _Colin Barker_, Jul 23 2018: (Start)

%F G.f.: x*(5 - 6*x + 4*x^2 - x^3)*(2 - 3*x + 6*x^2 - 4*x^3 + x^4) / (1 - x)^8.

%F a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) for n>8.

%F (End)

%e Some solutions for n=3:

%e ....0.2......0.1......0.2......0.2......0.1......0.0......0.1......0.1

%e ...2.2.2....1.2.2....1.2.2....1.2.2....0.2.2....0.0.2....0.1.2....0.2.2

%e ....3.3......3.3......3.3......2.3......3.3......1.3......1.2......2.2

%Y Cf. A216937.

%K nonn

%O 1,1

%A _R. H. Hardin_, Sep 20 2012

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