|
%I #47 Sep 08 2022 08:45:10
%S 1,4,9,18,35,68,133,262,519,1032,2057,4106,8203,16396,32781,65550,
%T 131087,262160,524305,1048594,2097171,4194324,8388629,16777238,
%U 33554455,67108888,134217753,268435482,536870939,1073741852,2147483677,4294967326,8589934623
%N a(n) = 2^(n+1)+n-1.
%C Is A247983(n+1) = A247983(n) if and only if n is in A083706? - _Clark Kimberling_, Sep 28 2014
%C a(n) is the least number of nodes in a height-n 2-3-4 tree, if using the top-down insertion algorithm and there have been no deletions. - _Daniel S. Roche_, Oct 05 2014
%C Also the number of independent vertex sets and vertex covers in the n-crown graph. - _Andrew Howroyd_, May 14 2017
%H Vincenzo Librandi, <a href="/A083706/b083706.txt">Table of n, a(n) for n = 0..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CrownGraph.html">Crown Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IndependentVertexSet.html">Independent Vertex Set</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/VertexCover.html">Vertex Cover</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,2).
%F G.f.: (1-2*x^2)/((1-x)^2*(1-2*x)).
%F a(n) = 2*a(n-1) + 3 - n.
%F Row sums of A130301. - _Gary W. Adamson_, May 20 2007
%t Table[2^(n + 1) + n - 1, {n, 0, 100}] (* _Vladimir Joseph Stephan Orlovsky_, Jul 19 2011 *)
%t LinearRecurrence[{4, -5, 2}, {4, 9, 18}, {0, 20}] (* _Eric W. Weisstein_, Sep 21 2017 *)
%t CoefficientList[Series[(-1 + 2 x^2)/((-1 + x)^2 (-1 + 2 x)), {x, 0, 20}], x] (* _Eric W. Weisstein_, Sep 21 2017 *)
%o (PARI) a(n)=if(n<0,0,2^(n+1)+n-1)
%o (Magma) [2^(n+1)+n-1: n in [0..35]]; // _Vincenzo Librandi_, Jul 20 2011
%Y Cf. A130301, A160692, A247983.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Jun 15 2003
|
