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A238190
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Number T(n,k) of equivalence classes of ways of placing k 3 X 3 tiles in an n X 4 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=3, 0<=k<=floor(n/3), read by rows.
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21
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1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 3, 4, 1, 3, 8, 1, 4, 12, 3, 1, 4, 18, 8, 1, 5, 24, 22, 1, 5, 32, 40, 6, 1, 6, 40, 73, 22, 1, 6, 50, 112, 66, 1, 7, 60, 172, 146, 10, 1, 7, 72, 240, 292, 48, 1, 8, 84, 335, 516, 174, 1, 8, 98, 440, 860, 448, 20
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OFFSET
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3,6
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LINKS
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EXAMPLE
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The first 13 rows of T(n,k) are:
.\ k 0 1 2 3 4 5
n
3 1 1
4 1 1
5 1 2
6 1 2 2
7 1 3 4
8 1 3 8
9 1 4 12 3
10 1 4 18 8
11 1 5 24 22
12 1 5 32 40 6
13 1 6 40 73 22
14 1 6 50 112 66
15 1 7 60 172 146 10
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MATHEMATICA
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T[n_, k_] := (2^k Binomial[n - 2k, k] + (Boole[EvenQ[k]] + Boole[OddQ[n] || EvenQ[k]] + Boole[k == 0]) 2^Quotient[k + 1, 2] Binomial[(n - 2k - Mod[n, 2])/2, Quotient[k, 2]])/4; Table[T[n, k], {n, 3, 20}, {k, 0, Floor[n/3]}] // Flatten (* Jean-François Alcover, Oct 06 2017, after Andrew Howroyd *)
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PROG
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(C++) See Gribble link.
(PARI)
T(n, k)={(2^k*binomial(n-2*k, k) + ((k%2==0)+(n%2==1||k%2==0)+(k==0)) * 2^((k+1)\2)*binomial((n-2*k-(n%2))/2, k\2))/4}
for(n=2, 20, for(k=0, floor(n/3), print1(T(n, k), ", ")); print) \\ Andrew Howroyd, May 29 2017
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CROSSREFS
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Cf. A034851, A226048, A102541, A226290, A238009, A228570, A225812, A238189, A228572, A228022, A231145, A231473, A231568, A232440, A228165, A238550, A238551, A238552, A228166, A238555, A238556, A228167, A238557, A238558, A238559, A228168, A238581, A238582, A238583, A228169, A238586, A238592.
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KEYWORD
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tabf,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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