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%I #15 May 30 2023 19:39:34
%S 3,13,40,127,430,1525,5572,20779,78682,301537,1166704,4548631,
%T 17840134,70297549,278001436,1102439683,4381060786,17438149561,
%U 69494317768,277202429935,1106485196638,4418967217573,17654948163700,70557030535387,282039835783690,1127594484061585
%N a(n) = 2*3^n + 4^n + 3*n.
%C For n >= 1, also the number of (non-null) connected induced subgraphs in the n-book graph.
%H Robert Israel, <a href="/A290720/b290720.txt">Table of n, a(n) for n = 0..1659</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BookGraph.html">Book Graph</a>
%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/Vertex-InducedSubgraph.html">Vertex-Induced Subgraph</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (9,-27,31,-12).
%F a(n) = 9*a(n-1) - 27*a(n-2) + 31*a(n-3) - 12*a(n-4).
%F G.f.: (3 - 14 x + 4 x^2 + 25 x^3)/((-1 + x)^2 (1 - 7 x + 12 x^2)).
%F From _Stefano Spezia_, May 30 2023: (Start)
%F a(n) = A008776(n) + A000302(n) + A008585(n).
%F E.g.f.: 2*exp(3*x) + exp(4*x) + 3*exp(x)*x. (End)
%p seq(2*3^n + 4^n + 3*n, n=0..30); # _Robert Israel_, Aug 09 2017
%t Table[2 3^n + 4^n + 3 n, {n, 0, 20}]
%t LinearRecurrence[{9, -27, 31, -12}, {13, 40, 127, 430}, {0, 20}]
%t CoefficientList[Series[(3 - 14 x + 4 x^2 + 25 x^3)/((-1 + x)^2 (1 - 7 x + 12 x^2)), {x, 0, 20}], x]
%Y Cf. A000302, A008585, A008776.
%K nonn,easy
%O 0,1
%A _Eric W. Weisstein_, Aug 09 2017
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