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A332599
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Triangle read by rows: T(n,k) = number of vertices in a "frame" of size n X k (see Comments in A331457 for definition).
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20
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5, 13, 37, 35, 99, 152, 75, 213, 256, 364, 159, 401, 448, 568, 776, 275, 657, 704, 836, 1056, 1340, 477, 1085, 1132, 1276, 1508, 1804, 2272, 755, 1619, 1712, 1868, 2112, 2420, 2900, 3532, 1163, 2327, 2552, 2720, 2976, 3296, 3788, 4432, 5336, 1659, 3257, 3568, 3748, 4016, 4348, 4852, 5508, 6424, 7516
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OFFSET
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1,1
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COMMENTS
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There is a crucial difference between frames of size nX2 and size nXk with k = 1 or k >= 3. If k != 2, all regions are either triangles or quadrilaterals, but for k=2 regions with larger numbers of sides can appear. Remember also that for k <= 2, the "frame" has no hole, and the graph has genus 0, whereas for k >= 3 there is a nontrivial hole and the graph has genus 1.
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LINKS
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FORMULA
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Column 1 is A331755, for which there is an explicit formula.
Column 2 is A331763, for which no formula is known.
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EXAMPLE
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Triangle begins:
[5],
[13, 37],
[35, 99, 152],
[75, 213, 256, 364],
[159, 401, 448, 568, 776],
[275, 657, 704, 836, 1056, 1340],
[477, 1085, 1132, 1276, 1508, 1804, 2272],
[755, 1619, 1712, 1868, 2112, 2420, 2900, 3532],
[1163, 2327, 2552, 2720, 2976, 3296, 3788, 4432, 5336],
[1659, 3257, 3568, 3748, 4016, 4348, 4852, 5508, 6424, 7516],
...
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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