A342939 a(n) is the Skolem number of the triangular grid graph T_n.

1, 2, 5, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, 172, 191, 211, 232, 254, 277, 301, 326, 352, 379, 407, 436, 466, 497, 529, 562, 596, 631, 667, 704, 742, 781, 821, 862, 904, 947, 991, 1036, 1082, 1129, 1177, 1226, 1276, 1327, 1379, 1432, 1486

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..55.

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 (3,-3,1).

Formula

O.g.f.: x*(1 - x + 2*x^2 - 3*x^3 + 3*x^4 - x^5)/(1 - x)^3.

E.g.f.: exp(x)*(2 + x^2)/2 - 1 + x^3/6.

a(n) = 3*a(n-1) - 3*a(n-2) - a(n-3) for n > 6.

Except for a(3) = 5:

a(n) = 1 + n*(n - 1)/2 (see Theorem 2.5 in Carrigan and Green).

a(n) = 1 + A161680(n).

a(n) = A152947(n-1).

Mathematica

LinearRecurrence[{3, -3, 1}, {1, 2, 5, 7, 11, 16}, 55]

Crossrefs

Cf. A152947, A161680, A247476, A342938, A342940.

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

Sequence in context: A317242 A217302 A062409 * A089781 A144832 A309290

Adjacent sequences: A342936 A342937 A342938 * A342940 A342941 A342942

Keyword

nonn,easy

Author

Stefano Spezia, Mar 30 2021

Status

approved