A290768 a(n) = (3/2)*(n^2 - n + 2).

3, 6, 12, 21, 33, 48, 66, 87, 111, 138, 168, 201, 237, 276, 318, 363, 411, 462, 516, 573, 633, 696, 762, 831, 903, 978, 1056, 1137, 1221, 1308, 1398, 1491, 1587, 1686, 1788, 1893, 2001, 2112, 2226, 2343, 2463, 2586, 2712, 2841, 2973, 3108, 3246, 3387, 3531, 3678

Offset: 1

Author

Eric W. Weisstein, Aug 10 2017

Keywords

• nonn
• easy

Comments

For n > 2, also the number of (non-null) connected induced subgraphs in the n-pan graph.

Links

• Eric Weisstein's World of Mathematics, Connected Graph.
• Eric Weisstein's World of Mathematics, Pan Graph.
• Eric Weisstein's World of Mathematics, Vertex-Induced Subgraph.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000124.

Programs

Mathematica

Table[3/2 (n^2 - n + 2), {n, 20}]
LinearRecurrence[{3, -3, 1}, {3, 6, 12}, 20]
CoefficientList[Series[-((3 (1 - x + x^2))/(-1 + x)^3), {x, 0, 20}], x]

Formula

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

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

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

a(n) = 3*A000124(n-1). - R. J. Mathar, May 07 2024

E.g.f.: -3 + 3*exp(x)*(1 + x^2/2). - Elmo R. Oliveira, May 31 2025