A342939 -id:A342939 - cp's OEIS frontend

A342938 a(n) is the Skolem number of the cycle graph C_n.

1, 1, 1, 3, 4, 4, 5, 4, 4, 5, 6, 7, 8, 8, 7, 8, 9, 9, 10, 11, 10, 12, 13, 12, 13, 13, 13, 15, 16, 16, 17, 16, 16, 17, 18, 19, 20, 20, 19, 20, 21, 21, 22, 23, 22, 24, 25, 24, 25, 25, 25, 27, 28, 28, 29, 28, 28, 29, 30, 31, 32, 32, 31, 32, 33, 33, 34, 35, 34, 36

Offset: 1

Stefano Spezia, Mar 30 2021

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

Braxton Carrigan and Garrett Green, Skolem Number of Subgraphs on the Triangular Lattice, Communications on Number Theory and Combinatorial Theory 2 (2021), Article 2.
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-2,2,-1,0,0,0,0,0,-1,2,-2,2,-2,2,-1).

Cf. A342939, A342940.

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

a[n_]:=If[MemberQ[{3,9,15,21},Mod[n,24]],(n-3)/2+1,If[MemberQ[{0,1,2,8,10,11,16,17,18,19},Mod[n,24]],Ceiling[n/2],Ceiling[n/2]+1]]; Array[a,70]

G.f.: x*(1 - x + x^2 + x^3 - x^7 + x^8 - 2*x^9 + 2*x^10 - x^11 + 2*x^12 - x^13 + 2*x^15 - x^16)/(1 - 2*x + 2*x^2 - 2*x^3 + 2*x^4 - 2*x^5 + x^6 + x^12 - 2*x^13 + 2*x^14 - 2*x^15 + 2*x^16 - 2*x^17 + x^18).
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - 2*a(n-4) + 2*a(n-5) - a(n-6) - a(n-12) + 2*a(n-13) - 2*a(n-14) + 2*a(n-15) - 2*a(n-16) + 2*a(n-17) - a(n-18) for n > 18.
a(n) = (n - 3)/2 + 1 when n is congruent to 3, 9, 15, 21 mod 24; ceiling(n/2) when n is congruent to 0, 1, 2, 8, 10, 11, 16, 17, 18, 19 mod 24; ceiling(n/2) + 1 when n is congruent to 4, 5, 6, 7, 12, 13, 14, 20, 22, 23 mod 24 (see Theorem 1.6 in Carrigan and Green).

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

Original entry on oeis.org

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

Stefano Spezia, Mar 30 2021

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

			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
...

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

Cf. A014206, A229183, A342938, A342939.

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

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

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).

cp's OEIS Frontend

A342938 a(n) is the Skolem number of the cycle graph C_n.

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