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A286191
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a(n) = (2^n-1)^2 + 2*n.
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3
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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
(list;
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refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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COMMENTS
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Number of connected induced (non-null) subgraphs of the complete bipartite graph K(n,n).
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LINKS
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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).
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FORMULA
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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)
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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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).
Sequence in context: A302757 A093834 A296045 * A033887 A291653 A183804
Adjacent sequences: A286188 A286189 A286190 * A286192 A286193 A286194
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KEYWORD
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nonn,easy
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AUTHOR
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Giovanni Resta, May 05 2017
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EXTENSIONS
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Name changed to the formula by Eric W. Weisstein, Aug 09 2017
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STATUS
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approved
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