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a(n) = 3/2*(n^2 - n + 2).
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%I #8 May 07 2024 06:57:50

%S 3,6,12,21,33,48,66,87,111,138,168,201,237,276,318,363,411,462,516,

%T 573,633,696,762,831,903,978,1056,1137,1221,1308,1398,1491,1587,1686,

%U 1788,1893,2001,2112,2226,2343,2463,2586,2712,2841,2973,3108,3246,3387,3531,3678

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

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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ConnectedGraph.html">Connected Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PanGraph.html">Pan Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Vertex-InducedSubgraph.html">Vertex-Induced Subgraph</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3, -3, 1).

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

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

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

%F a(n) = 3*A000124(n-1). - _R. J. Mathar_, May 07 2024

%t Table[3/2 (n^2 - n + 2), {n, 20}]

%t LinearRecurrence[{3, -3, 1}, {3, 6, 12}, 20]

%t CoefficientList[Series[-((3 (1 - x + x^2))/(-1 + x)^3), {x, 0, 20}], x]

%K nonn,easy

%O 1,1

%A _Eric W. Weisstein_, Aug 10 2017