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

%I A342940 #9 Apr 01 2021 14:56:27
%S A342940 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 A342940 38,44,9,16,23,30,37,44,51,58,10,18,26,34,42,50,58,66,74,11,20,29,38,
%U A342940 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 A342940 Triangle read by rows: T(n, k) is the Skolem number of the parallelogram graph P_{n, k}, with 1 < k <= n.
%C A342940 For the meaning of Skolem number of a graph, see Definitions 1.4 and 1.5 in Carrigan and Green.
%H A342940 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 A342940 O.g.f.: (4 - 6*y - x*(5 - 8*y))/((1 - x)^2*(1 - y)^2).
%F A342940 E.g.f.: exp(x+y)*(4 - x*(1 - y) - 2*y).
%F A342940 T(n, k) = k*n - 2*k - n + 4 (see Theorem 3.3 in Carrigan and Green).
%F A342940 Sum_{k=2..n} T(n, k) = A229183(n-1).
%F A342940 T(n, n) = A014206(n-2).
%e A342940 The triangle T(n, k) begins:
%e A342940 n\k|   2   3   4   5   6   7
%e A342940 ---+------------------------
%e A342940 2  |   2
%e A342940 3  |   3   4
%e A342940 4  |   4   6   8
%e A342940 5  |   5   8  11  14
%e A342940 6  |   6  10  14  18  22
%e A342940 7  |   7  12  17  22  27  32
%e A342940 ...
%t A342940 T[n_,k_]:=k*n-2k-n+4; Table[T[n,k],{n,2,13},{k,2,n}]//Flatten
%Y A342940 Cf. A014206, A229183, A342938, A342939.
%Y A342940 For n > 1, 3*A002061(n) gives the Skolem number of the hexagonal grid graph H_n.
%K A342940 nonn,easy,tabl
%O A342940 2,1
%A A342940 _Stefano Spezia_, Mar 30 2021