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Number of n X 2 arrays of the minimum value of corresponding elements and their horizontal, vertical or antidiagonal neighbors in a random, but sorted with lexicographically nondecreasing rows and columns, 0..3 n X 2 array.
1

%I #9 Aug 03 2018 08:11:59

%S 4,13,41,114,260,545,1104,2182,4201,7862,14295,25272,43500,73013,

%T 119684,191880,301285,463918,701375,1042326,1524300,2195793,3118736,

%U 4371362,6051513,8280430,11207071,15013004,19917924,26185845,34132020,44130644

%N Number of n X 2 arrays of the minimum value of corresponding elements and their horizontal, vertical or antidiagonal neighbors in a random, but sorted with lexicographically nondecreasing rows and columns, 0..3 n X 2 array.

%C Column 2 of A220931.

%H R. H. Hardin, <a href="/A220926/b220926.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = (1/20160)*n^8 - (1/2520)*n^7 - (1/1440)*n^6 + (7/45)*n^5 - (3953/2880)*n^4 + (2093/360)*n^3 + (1049/560)*n^2 - (12377/420)*n + 28 for n>3.

%F Conjectures from _Colin Barker_, Aug 03 2018: (Start)

%F G.f.: x*(4 - 23*x + 68*x^2 - 123*x^3 + 122*x^4 - x^5 - 153*x^6 + 172*x^7 - 65*x^8- 8*x^9 + 11*x^10 - 2*x^11) / (1 - x)^9.

%F a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) +36*a(n-7) - 9*a(n-8) + a(n-9) for n>12.

%F (End)

%e Some solutions for n=3:

%e ..1..1....1..1....0..0....1..1....0..0....0..0....1..1....0..0....1..1....1..1

%e ..1..1....1..1....0..2....1..1....0..1....0..2....1..1....0..1....1..2....1..1

%e ..3..3....1..1....2..3....2..3....1..3....3..3....1..2....1..2....3..3....1..3

%Y Cf. A220931.

%K nonn

%O 1,1

%A _R. H. Hardin_, Dec 25 2012