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%I #44 Nov 09 2023 08:50:56
%S 1,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,
%T 50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,
%U 96,98,100,102,104,106,108,110,112,114,116,118,120,122,124,126
%N Expansion of (1+2*x-x^2)/(1-x)^2.
%C Row sums of A103516.
%C Also the number of maximal and maximum cliques in the (n+1) X (n+1) rook graph. - _Eric W. Weisstein_, Sep 14 2017
%C Also the number of maximal and maximum independent vertex sets in the (n+1) X (n+1) rook complement graph. - _Eric W. Weisstein_, Sep 14 2017
%H G. C. Greubel, <a href="/A103517/b103517.txt">Table of n, a(n) for n = 0..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MaximalClique.html">Maximal Clique</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MaximumClique.html">Maximum Clique</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MaximalIndependentVertexSet.html">Maximal Independent Vertex Set</a>
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MaximumIndependentVertexSet.html">Maximum Independent Vertex Set</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RookComplementGraph.html">Rook Complement Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RookGraph.html">Rook Graph</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F a(n) = 2*n + 2 - 0^n.
%F a(n) = Sum_{k=0..n} 0^(k(n-k))*(n+1).
%F Equals binomial transform of [1, 3, -1, 1, -1, 1, ...]. - _Gary W. Adamson_, Apr 23 2008
%F a(n) = 2*a(n-1) - a(n-2) for n > 2. - _Eric W. Weisstein_, Sep 14 2017
%F G.f.: (1 + 2*x - x^2)/(-1 + x)^2. - _Eric W. Weisstein_, Sep 14 2017
%p a := n -> 2*(n + 1) - 0^n: seq(a(n), n = 0..62); # _Peter Luschny_, May 12 2023
%t CoefficientList[Series[(-z^2 + 2*z + 1)/(z - 1)^2, {z, 0, 100}], z] (* _Vladimir Joseph Stephan Orlovsky_, Jul 10 2011 *)
%t CoefficientList[Series[2/(x - 1)^2 - 1, {x, 0, 62}], x] (* _Robert G. Wilson v_, Jan 29 2015 *)
%t Join[{1}, 2 Range[2, 20]] (* _Eric W. Weisstein_, Sep 14 2017 *)
%t Join[{1}, LinearRecurrence[{2, -1}, {4, 6}, 20]] (* _Eric W. Weisstein_, Sep 14 2017 *)
%o (Magma) [1] cat [2*n+2 : n in [1..60]]; // _Wesley Ivan Hurt_, Dec 07 2016
%Y Cf. A103516.
%Y Essentially the same as A004277, A005843, A051755, and A076032. - _R. J. Mathar_, Jul 31 2010
%Y Cf. A272651 (for which this sequence is a conjectured continuation for large n).
%K easy,nonn
%O 0,2
%A _Paul Barry_, Feb 09 2005
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