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A085279 Expansion of (1 - 2*x - 2*x^2)/((1 - 2*x)*(1 - 3*x)).

%I A085279 #30 Feb 16 2025 08:32:50
%S A085279 1,3,7,17,43,113,307,857,2443,7073,20707,61097,181243,539633,1610707,
%T A085279 4815737,14414443,43177793,129402307,387944777,1163310043,3488881553,
%U A085279 10464547507,31389448217,94159956043,282463090913,847355718307
%N A085279 Expansion of (1 - 2*x - 2*x^2)/((1 - 2*x)*(1 - 3*x)).
%C A085279 Binomial transform of A001045(n)+1.
%C A085279 For n > 1, also the number of independent vertex sets in the (n-1)-book graph. - _Eric W. Weisstein_, Aug 16 2017
%H A085279 Vincenzo Librandi, <a href="/A085279/b085279.txt">Table of n, a(n) for n = 0..1000</a>
%H A085279 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BookGraph.html">Book Graph</a>
%H A085279 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IndependentVertexSet.html">Independent Vertex Set</a>
%H A085279 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-6).
%F A085279 a(n) = (3*2^n + 3^n - 0^n)/3.
%F A085279 a(n) = 2^n + 3^(n-1) for n >= 1.
%F A085279 G.f.: (1 - 2*x - 2*x*x)/((1 - 2*x)*(1 - 3*x)).
%F A085279 a(n) = 5*a(n-1) - 6*a(n-2) for n > 1. - _Vincenzo Librandi_, Sep 12 2014
%F A085279 E.g.f.: (1/3)*(exp(3*x) + 3*exp(2*x) -1). - _G. C. Greubel_, Aug 17 2017
%p A085279 seq(2^n + (3^n - charfcn[0](n))/3, n=0..100); # _Robert Israel_, Sep 12 2014
%t A085279 CoefficientList[Series[(1 - 2 x - 2 x^2)/((1 - 2 x) (1 - 3 x)), {x, 0, 30}], x] (* _Vincenzo Librandi_, Sep 12 2014 *)
%t A085279 Join[{1}, LinearRecurrence[{5, -6}, {3, 7}, 20]] (* _Eric W. Weisstein_, Aug 16 2017 *)
%t A085279 Join[{1}, Table[2^n + 3^(n - 1), {n, 20}]] (* _Eric W. Weisstein_, Aug 16 2017 *)
%o A085279 (Magma) [1] cat [2^n+3^(n-1): n in [1..30]]; // _Vincenzo Librandi_, Sep 12 2014
%o A085279 (PARI) Vec((1-2*x-2*x*x)/((1-2*x)*(1-3*x)) + O(x^50)) \\ _Michel Marcus_, Sep 12 2014
%Y A085279 Cf. A001045.
%K A085279 easy,nonn
%O A085279 0,2
%A A085279 _Paul Barry_, Jun 25 2003