A287988 -id:A287988 - cp's OEIS frontend

A338154 a(n) is the number of acyclic orientations of the edges of the n-antiprism.

426, 4968, 50640, 486930, 4547088, 41796168, 380789562, 3451622904, 31194607488, 281440825122, 2536622917920, 22848990484344, 205743704494026, 1852238413383048, 16673036119790640, 150072652217086770, 1350735146332489008, 12157047307392618408

Offset: 3

Peter Kagey, Oct 13 2020

nonn

Conjectured linear recurrence and g.f. confirmed by Kagey's formula. - Ray Chandler, Mar 10 2024

			For n = 3, the 3-antiprism is the octahedron (3-dimensional cross-polytope), so a(3) = A033815(3) = 426.

Peter Kagey, Table of n, a(n) for n = 3..1000
Eric Weisstein's World of Mathematics, Antiprism Graph
Wikipedia, Acyclic orientation
Index entries for linear recurrences with constant coefficients, signature (17, -88, 153, -81).

Cf. A077263, A124352, A124353, A192742, A284699, A284700, A284701, A287988, A294152, A297384.

Cf. A033815 (cross-polytope), A058809 (wheel), A334247 (hypercube), A338152 (demihypercube), A338153 (prism).

A338154[n_] := Round[-2^(1-n)*((7 - Sqrt[13])^n + (7 + Sqrt[13])^n) + 9^n + 5] (* Peter Kagey, Nov 15 2020 *)

Conjectures from Colin Barker, Oct 13 2020: (Start)
G.f.: 6*x^3*(71 - 379*x + 612*x^2 - 324*x^3) / ((1 - x)*(1 - 9*x)*(1 - 7*x + 9*x^2)).
a(n) = 17*a(n-1) - 88*a(n-2) + 153*a(n-3) - 81*a(n-4) for n>6.
(End)
a(n) = -2^(1-n)*((7-sqrt(13))^n + (7+sqrt(13))^n) + 9^n + 5. - Peter Kagey, Nov 15 2020

A287992 Number of (undirected) paths in the prism graph Y_n.

Original entry on oeis.org

1, 26, 129, 444, 1285, 3366, 8281, 19544, 44829, 100770, 223201, 488916, 1061749, 2289854, 4910505, 10480176, 22275661, 47178234, 99605809, 209704940, 440390181, 922733526, 1929364729, 4026514824, 8388588925, 17448283346, 36238762881, 75161901444, 155692535509, 322122515310

Offset: 1

Eric W. Weisstein, Jun 04 2017

Extended to a(1)-a(2) using the formula.

Colin Barker, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Graph Path.
Eric Weisstein's World of Mathematics, Prism Graph.
Index entries for linear recurrences with constant coefficients, signature (8,-26,44,-41,20,-4).

Cf. A077265, A124350, A124349, A287988, A020874, A288516.

Table[(5 2^(n + 1) - 5 n - n^2 - 13) n, {n, 20}]
LinearRecurrence[{8, -26, 44, -41, 20, -4}, {1, 26, 129, 444, 1285, 3366}, 20]
CoefficientList[Series[(1 + 18 x - 53 x^2 + 44 x^3 - 16 x^4)/((1 - x)^4 (1 - 2 x)^2), {x, 0, 20}], x]

Vec(x*(1 + 18*x - 53*x^2 + 44*x^3 - 16*x^4) / ((1 - x)^4*(1 - 2*x)^2) + O(x^30)) \\ Colin Barker, Jun 04 2017

a(n) = (5*2^(n + 1) - 5*n - n^2 - 13)*n.
From Colin Barker, Jun 04 2017: (Start)
G.f.: x*(1 + 18*x - 53*x^2 + 44*x^3 - 16*x^4) / ((1 - x)^4*(1 - 2*x)^2).
a(n) = 8*a(n-1) - 26*a(n-2) + 44*a(n-3) - 41*a(n-4) + 20*a(n-5) - 4*a(n-6) for n>6. (End)

cp's OEIS Frontend

A338154 a(n) is the number of acyclic orientations of the edges of the n-antiprism.

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