This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A083706 #51 Mar 06 2025 20:32:59
%S A083706 1,4,9,18,35,68,133,262,519,1032,2057,4106,8203,16396,32781,65550,
%T A083706 131087,262160,524305,1048594,2097171,4194324,8388629,16777238,
%U A083706 33554455,67108888,134217753,268435482,536870939,1073741852,2147483677,4294967326,8589934623,17179869216
%N A083706 a(n) = 2^(n+1) + n - 1.
%C A083706 Is A247983(n+1) = A247983(n) if and only if n is in A083706? - _Clark Kimberling_, Sep 28 2014
%C A083706 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 A083706 Also the number of independent vertex sets and vertex covers in the n-crown graph. - _Andrew Howroyd_, May 14 2017
%H A083706 Vincenzo Librandi, <a href="/A083706/b083706.txt">Table of n, a(n) for n = 0..1000</a>
%H A083706 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CrownGraph.html">Crown Graph</a>.
%H A083706 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IndependentVertexSet.html">Independent Vertex Set</a>.
%H A083706 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/VertexCover.html">Vertex Cover</a>.
%H A083706 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,2).
%F A083706 G.f.: (1-2*x^2)/((1-x)^2*(1-2*x)).
%F A083706 a(n) = 2*a(n-1) + 3 - n.
%F A083706 Row sums of A130301. - _Gary W. Adamson_, May 20 2007
%F A083706 From _Elmo R. Oliveira_, Mar 06 2025: (Start)
%F A083706 E.g.f.: exp(x)*(x + 2*exp(x) - 1).
%F A083706 a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3). (End)
%t A083706 Table[2^(n + 1) + n - 1, {n, 0, 100}] (* _Vladimir Joseph Stephan Orlovsky_, Jul 19 2011 *)
%t A083706 LinearRecurrence[{4, -5, 2}, {4, 9, 18}, {0, 20}] (* _Eric W. Weisstein_, Sep 21 2017 *)
%t A083706 CoefficientList[Series[(-1 + 2 x^2)/((-1 + x)^2 (-1 + 2 x)), {x, 0, 20}], x] (* _Eric W. Weisstein_, Sep 21 2017 *)
%o A083706 (PARI) a(n)=if(n<0,0,2^(n+1)+n-1)
%o A083706 (Magma) [2^(n+1)+n-1: n in [0..35]]; // _Vincenzo Librandi_, Jul 20 2011
%Y A083706 Cf. A083706, A130301, A160692, A247983.
%K A083706 nonn,easy
%O A083706 0,2
%A A083706 _N. J. A. Sloane_, Jun 15 2003