A342940 Triangle read by rows: T(n, k) is the Skolem number of the parallelogram graph P_{n, k}, with 1 < k <= n.

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

Offset: 2

Author

Stefano Spezia, Mar 30 2021

Keywords

• nonn
• easy
• tabl

Comments

For the meaning of Skolem number of a graph, see Definitions 1.4 and 1.5 in Carrigan and Green.

Examples

			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
...
		

Links

• Braxton Carrigan and Garrett Green, Skolem Number of Subgraphs on the Triangular Lattice, Communications on Number Theory and Combinatorial Theory 2 (2021), Article 2.

Crossrefs

Cf. A014206, A229183, A342938, A342939.

For n > 1, 3*A002061(n) gives the Skolem number of the hexagonal grid graph H_n.

Programs

Mathematica

T[n_,k_]:=k*n-2k-n+4; Table[T[n,k],{n,2,13},{k,2,n}]//Flatten

Formula

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).

T(n, n) = A014206(n-2).