A342940 - OEIS

%I #9 Apr 01 2021 14:56:27

%S 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,

%T 38,44,9,16,23,30,37,44,51,58,10,18,26,34,42,50,58,66,74,11,20,29,38,

%U 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

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

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

%H Braxton Carrigan and Garrett Green, <a href="https://research.library.kutztown.edu/contact/vol2/iss1/2/">Skolem Number of Subgraphs on the Triangular Lattice</a>, Communications on Number Theory and Combinatorial Theory 2 (2021), Article 2.

%F O.g.f.: (4 - 6*y - x*(5 - 8*y))/((1 - x)^2*(1 - y)^2).

%F E.g.f.: exp(x+y)*(4 - x*(1 - y) - 2*y).

%F T(n, k) = k*n - 2*k - n + 4 (see Theorem 3.3 in Carrigan and Green).

%F Sum_{k=2..n} T(n, k) = A229183(n-1).

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

%e The triangle T(n, k) begins:

%e n\k|   2   3   4   5   6   7

%e ---+------------------------

%e 2  |   2

%e 3  |   3   4

%e 4  |   4   6   8

%e 5  |   5   8  11  14

%e 6  |   6  10  14  18  22

%e 7  |   7  12  17  22  27  32

%e ...

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

%Y Cf. A014206, A229183, A342938, A342939.

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

%K nonn,easy,tabl

%O 2,1

%A _Stefano Spezia_, Mar 30 2021