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Number of n X 2 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.

2

`%I #17 Mar 18 2024 10:52:47
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`%S 1,3,5,4,12,16,9,27,33,16,48,56,25,75,85,36,108,120,49,147,161,64,192,
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`%T 208,81,243,261,100,300,320,121,363,385,144,432,456,169,507,533,196,
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`%U 588,616,225,675,705,256,768,800,289,867,901,324,972,1008,361,1083,1121,400,1200
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`%N Number of n X 2 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.
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`%H Alois P. Heinz, <a href="/A201271/b201271.txt">Table of n, a(n) for n = 0..10000</a> (terms n = 1..210 from R. H. Hardin)
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`%F a(n) = 3*a(n-3) -3*a(n-6) +a(n-9).
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`%F Subsequences for n modulo 3 = 1,2,0:
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`%F p=(n+2)/3: a(n) = 3*p^2
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`%F q=(n+1)/3: a(n) = 3*q^2 + 2*q
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`%F r=(n+0)/3: a(n) = r^2 + 2*r + 1.
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`%F G.f.: 1+x*(3 + 5*x + 4*x^2 + 3*x^3 + x^4 - 3*x^5 + x^8) / ((1 - x)^3*(1 + x + x^2)^3). - _Colin Barker_, May 22 2018
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`%e Some solutions for n=5:
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`%e ..0..1....0..1....0..0....0..0....0..0....0..0....0..0....0..1....0..1....0..0
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`%e ..0..1....0..1....0..0....0..1....0..1....0..1....0..0....0..2....0..1....0..2
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`%e ..0..1....0..1....1..2....1..1....1..1....1..2....1..1....0..2....0..2....1..2
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`%e ..0..2....1..2....1..2....2..2....1..2....1..2....1..2....1..2....1..2....1..2
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`%e ..2..2....2..2....1..2....2..2....2..2....2..2....2..2....1..2....2..2....1..2
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`%Y Column 2 of A201277.
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`%K nonn,easy
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`%O 0,2
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`%A _R. H. Hardin_, Nov 29 2011
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`%E a(0)=1 prepended by _Alois P. Heinz_, Mar 18 2024
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