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A342940
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Triangle read by rows: T(n, k) is the Skolem number of the parallelogram graph P_{n, k}, with 1 < k <= n.
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2
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2, 3, 4, 4, 6, 8, 5, 8, 11, 14, 6, 10, 14, 18, 22, 7, 12, 17, 22, 27, 32, 8, 14, 20, 26, 32, 38, 44, 9, 16, 23, 30, 37, 44, 51, 58, 10, 18, 26, 34, 42, 50, 58, 66, 74, 11, 20, 29, 38, 47, 56, 65, 74, 83, 92, 12, 22, 32, 42, 52, 62, 72, 82, 92, 102, 112, 13, 24, 35, 46, 57, 68, 79, 90, 101, 112, 123, 134
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OFFSET
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2,1
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COMMENTS
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For the meaning of Skolem number of a graph, see Definitions 1.4 and 1.5 in Carrigan and Green.
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LINKS
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FORMULA
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O.g.f.: (4 - 6*y - x*(5 - 8*y))/((1 - x)^2*(1 - y)^2).
E.g.f.: exp(x+y)*(4 - x*(1 - y) - 2*y).
T(n, k) = k*n - 2*k - n + 4 (see Theorem 3.3 in Carrigan and Green).
Sum_{k=2..n} T(n, k) = A229183(n-1).
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EXAMPLE
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The triangle T(n, k) begins:
n\k| 2 3 4 5 6 7
---+------------------------
2 | 2
3 | 3 4
4 | 4 6 8
5 | 5 8 11 14
6 | 6 10 14 18 22
7 | 7 12 17 22 27 32
...
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MATHEMATICA
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T[n_, k_]:=k*n-2k-n+4; Table[T[n, k], {n, 2, 13}, {k, 2, n}]//Flatten
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CROSSREFS
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For n > 1, 3*A002061(n) gives the Skolem number of the hexagonal grid graph H_n.
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KEYWORD
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AUTHOR
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STATUS
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approved
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