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 A286191 #12 Feb 16 2025 08:33:44
%S A286191 3,13,55,233,971,3981,16143,65041,261139,1046549,4190231,16769049,
%T A286191 67092507,268402717,1073676319,4294836257,17179607075,68718952485,
%U A286191 274876858407,1099509530665,4398042316843,17592177655853,70368727400495,281474943156273,1125899839733811
%N A286191 a(n) = (2^n-1)^2 + 2*n.
%C A286191 Number of connected induced (non-null) subgraphs of the complete bipartite graph K(n,n).
%H A286191 Colin Barker, <a href="/A286191/b286191.txt">Table of n, a(n) for n = 1..1000</a>
%H A286191 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>
%H A286191 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Vertex-InducedSubgraph.html">Vertex-Induced Subgraph</a>
%H A286191 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-21,22,-8).
%F A286191 a(n) = (2^n-1)^2 + 2*n.
%F A286191 From _Colin Barker_, May 30 2017: (Start)
%F A286191 G.f.: x*(3 - 11*x + 14*x^2) / ((1 - x)^2*(1 - 2*x)*(1 - 4*x)).
%F A286191 a(n) = 8*a(n-1) - 21*a(n-2) + 22*a(n-3) - 8*a(n-4) for n>4.
%F A286191 (End)
%t A286191 a[n_] := (2^n-1)^2 + 2*n; Array[a, 30]
%t A286191 Table[(2^n - 1)^2 + 2 n, {n, 20}] (* _Eric W. Weisstein_, Aug 09 2017 *)
%t A286191 LinearRecurrence[{8, -21, 22, -8}, {3, 13, 55, 233}, 20] (* _Eric W. Weisstein_, Aug 09 2017 *)
%t A286191 CoefficientList[Series[(3 - 11 x + 14 x^2)/((-1 + x)^2 (1 - 6 x + 8 x^2)), {x, 0, 20}], x] (* _Eric W. Weisstein_, Aug 09 2017 *)
%o A286191 (PARI) Vec(x*(3 - 11*x + 14*x^2) / ((1 - x)^2*(1 - 2*x)*(1 - 4*x)) + O(x^30)) \\ _Colin Barker_, May 30 2017
%Y A286191 Cf. A020873 (wheel), A059020 (ladder), A059525 (grid), A286139 (king), A286182 (prism), A286183 (antiprism), A286184 (helm), A286185 (Möbius ladder), A286186 (friendship), A286187 (web), A286188 (gear), A286189 (rook), A285765 (queen).
%K A286191 nonn,easy
%O A286191 1,1
%A A286191 _Giovanni Resta_, May 05 2017
%E A286191 Name changed to the formula by _Eric W. Weisstein_, Aug 09 2017