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Number of n X 3 0..2 arrays with every row and column nondecreasing rightwards and downwards, and the number of instances of each value within one of each other.
3

%I #17 Mar 18 2024 09:02:17

%S 1,1,4,7,14,21,41,54,86,120,168,218,307,377,496,621,776,937,1177,1380,

%T 1676,1984,2344,2716,3221,3665,4260,4875,5570,6285,7201,8026,9074,

%U 10152,11344,12566,14071,15449,17136,18865,20748,22673,24977,27112,29656,32256,35056

%N Number of n X 3 0..2 arrays with every row and column nondecreasing rightwards and downwards, and the number of instances of each value within one of each other.

%H Alois P. Heinz, <a href="/A201272/b201272.txt">Table of n, a(n) for n = 0..10000</a> (terms n = 1..210 from R. H. Hardin)

%F a(n) = 2*a(n-2) + 2*a(n-3) - a(n-4) - 4*a(n-5) + 2*a(n-7) - 2*a(n-9) + 4*a(n-11) + a(n-12) - 2*a(n-13) - 2*a(n-14) + a(n-16).

%F G.f.: 1 + x*(1 + 4*x + 5*x^2 + 4*x^3 + 7*x^5 + 7*x^6 + 2*x^7 - x^8 + x^9 + 4*x^10 + x^11 - 2*x^12 - 2*x^13 + x^15) / ((1 - x)^5*(1 + x)^3*(1 - x + x^2)*(1 + x + x^2)^3). - _Colin Barker_, Mar 02 2018

%e Some solutions for n=5:

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

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

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

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

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

%Y Column 3 of A201277.

%K nonn,easy

%O 0,3

%A _R. H. Hardin_, Nov 29 2011

%E a(0)=1 prepended by _Alois P. Heinz_, Mar 18 2024