A286191 a(n) = (2^n-1)^2 + 2*n.

3, 13, 55, 233, 971, 3981, 16143, 65041, 261139, 1046549, 4190231, 16769049, 67092507, 268402717, 1073676319, 4294836257, 17179607075, 68718952485, 274876858407, 1099509530665, 4398042316843, 17592177655853, 70368727400495, 281474943156273, 1125899839733811

Offset: 1

Author

Giovanni Resta, May 05 2017

Keywords

• nonn
• easy

Comments

Number of connected induced (non-null) subgraphs of the complete bipartite graph K(n,n).

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Complete Bipartite Graph
• Eric Weisstein's World of Mathematics, Vertex-Induced Subgraph
• Index entries for linear recurrences with constant coefficients, signature (8,-21,22,-8).

Crossrefs

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).

Programs

Mathematica

a[n_] := (2^n-1)^2 + 2*n; Array[a, 30]
Table[(2^n - 1)^2 + 2 n, {n, 20}] (* Eric W. Weisstein, Aug 09 2017 *)
LinearRecurrence[{8, -21, 22, -8}, {3, 13, 55, 233}, 20] (* Eric W. Weisstein, Aug 09 2017 *)
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 *)

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

Formula

a(n) = (2^n-1)^2 + 2*n.

From Colin Barker, May 30 2017: (Start)

G.f.: x*(3 - 11*x + 14*x^2) / ((1 - x)^2*(1 - 2*x)*(1 - 4*x)).

a(n) = 8*a(n-1) - 21*a(n-2) + 22*a(n-3) - 8*a(n-4) for n>4.

(End)

Extensions

Name changed to the formula by Eric W. Weisstein, Aug 09 2017