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A324009
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The number of convex polyominoes whose smallest bounding rectangle has size w*h (w > 0, h > 0). The table is read by antidiagonals.
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1
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1, 1, 1, 1, 5, 1, 1, 13, 13, 1, 1, 25, 68, 25, 1, 1, 41, 222, 222, 41, 1, 1, 61, 555, 1110, 555, 61, 1, 1, 85, 1171, 3951, 3951, 1171, 85, 1, 1, 113, 2198, 11263, 19010, 11263, 2198, 113, 1, 1, 145, 3788, 27468, 70438, 70438, 27468, 3788, 145, 1
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OFFSET
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1,5
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LINKS
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FORMULA
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a(w,h) = binomial(2w+2h-4, 2w-2) + ((2w+2h-5)/2)*binomial(2w+2h-6, 2w-3) - 2(w+h-3)*binomial(w+h-2, w-1)*binomial(w+h-4, w-2), for w > 0, h > 0.
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EXAMPLE
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For w=3 and h=2, the a(3,2)=13 polyominoes spanning a 3 X 2 rectangle are
XXX X XX X XX
XXX XXX XX XXX XXX
plus all different horizontal and vertical reflections (1+2+2+4+4=13).
Table begins
1 1 1 1 1 1 1 ...
1 5 13 25 41 61 ...
1 13 68 222 555 ...
1 25 222 1110 ...
1 41 555 ...
1 61 ...
1 ...
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MATHEMATICA
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Table[Binomial[2 # + 2 h - 4, 2 # - 2] + ((2 # + 2 h - 5)/2) Binomial[2 # + 2 h - 6, 2 # - 3] - 2 (# + h - 3) Binomial[# + h - 2, # - 1] Binomial[# + h - 4, # - 2] &[w - h + 1], {w, 10}, {h, w}] // Flatten (* Michael De Vlieger, Apr 15 2019 *)
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PROG
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(Sage)
def a(w, h):
s = w+h # half the perimeter
return ( binomial(2*s-4, 2*w-2) + binomial(2*s-6, 2*w-3)*(s-5/2)
- 2*(s-3)*binomial(s-2, w-1)*binomial(s-4, w-2) )
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CROSSREFS
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A093118 contains the same data in a different arrangement and without the entries for w=1 and for h=1.
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KEYWORD
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AUTHOR
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STATUS
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approved
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