A290764 Number of (non-null) connected induced subgraphs in the 2 X n king graph.

3, 15, 54, 174, 537, 1629, 4908, 14748, 44271, 132843, 398562, 1195722, 3587205, 10761657, 32285016, 96855096, 290565339, 871696071, 2615088270, 7845264870, 23535794673, 70607384085, 211822152324, 635466457044, 1906399371207, 5719198113699, 17157594341178

Offset: 1

Author

Eric W. Weisstein, Aug 10 2017

Keywords

• nonn
• easy

Comments

a(n) is also the number of 4-cycles in the (n+1)-Dorogovtsev-Goltsev-Mendes graph (using the indexing convention that the 0-Dorogovtsev-Goltsev-Mendes graph is P_2). - Eric W. Weisstein, Dec 06 2023

Links

• Eric Weisstein's World of Mathematics, Connected Graph.
• Eric Weisstein's World of Mathematics, Dorogovtsev-Goltsev-Mendes Graph.
• Eric Weisstein's World of Mathematics, Graph Cycle.
• Eric Weisstein's World of Mathematics, King Graph.
• Eric Weisstein's World of Mathematics, Vertex-Induced Subgraph.
• Index entries for linear recurrences with constant coefficients, signature (5, -7, 3).

Crossrefs

Cf. A003462(n) (3-cycles), A367967(n) (5-cycles), A367968(n) (6-cycles).

Programs

Mathematica

Table[3/4 (3^(n + 1) - 2 n - 3), {n, 20}]
LinearRecurrence[{5, -7, 3}, {3, 15, 54}, 40]
CoefficientList[Series[-3/((-1 + x)^2 (-1 + 3 x)), {x, 0, 20}], x]

Formula

a(n) = 3/4*(3^(n + 1) - 2*n - 3).

a(n) = 5*a(n-1) - 7*a(n-2) + 3*a(n-3).

G.f.: -((3 x)/((-1 + x)^2 (-1 + 3 x))).