A286182 Number of connected induced (non-null) subgraphs of the prism graph with 2n nodes.

3, 13, 51, 167, 503, 1441, 4007, 10923, 29355, 78037, 205659, 538127, 1399583, 3621289, 9327695, 23931603, 61186131, 155949085, 396369795, 1004904695, 2541896519, 6416348209, 16165610999, 40657256571, 102090514683, 255968753125, 640899345579, 1602640560479

Offset

1,1

Comments

Cases n=1 and n=2 correspond to degenerate prism graphs, but they fit the same (conjectured) linear recurrence as the other terms.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

Eric Weisstein's World of Mathematics, Prism Graph

Eric Weisstein's World of Mathematics, Vertex-Induced Subgraph

Formula

a(n) = 6*a(n-1) - 11*a(n-2) + 4*a(n-3) + 5*a(n-4) - 2*a(n-5) - a(n-6), for n > 6 (conjectured).

a(n) = A002203(n) + 3*n*A000129(n) - 3*n + 1 (conjectured). - Eric W. Weisstein, May 08 2017

G.f.: x*(3 - 5*x + 6*x^2 - 8*x^3 - 5*x^4 - 3*x^5) / ((1 - x)^2*(1 - 2*x - x^2)^2) (conjectured). - Colin Barker, May 31 2017

Mathematica

a[n_] := Block[{g = Graph@ Flatten@ Table[{i <-> Mod[i, n] + 1, n+i <-> Mod[i, n] + n+1, i <-> i+n}, {i, n}]}, -1 + ParallelSum[ Boole@ ConnectedGraphQ@ Subgraph[g, s], {s, Subsets@Range[2 n]}]]; Array[a, 8]

Crossrefs

Cf. A020873 (wheel), A059020 (ladder), A059525 (grid), A286139 (king), A286183 (antiprism), A286184 (helm), A286185 (Möbius ladder), A286186 (friendship), A286187 (web), A286188 (gear), A286189 (rook), A285765 (queen).

Sequence in context: A026529 A363881 A357870 * A101052 A016064 A163774

Adjacent sequences: A286179 A286180 A286181 * A286183 A286184 A286185

Keyword

nonn

Author

Giovanni Resta, May 04 2017

Extensions

Terms a(18) and beyond from Andrew Howroyd, Aug 15 2017

Status

approved