A124352 -id:A124352 - cp's OEIS frontend

A124353 Number of (directed) Hamiltonian circuits on the n-antiprism graph.

6, 18, 32, 58, 112, 220, 450, 938, 1982, 4220, 9022, 19332, 41472, 89022, 191150, 410506, 881656, 1893634, 4067256, 8735972, 18763898, 40302866, 86566390, 185935764, 399371142, 857808780, 1842486536, 3957474934, 8500256470, 18257692546, 39215680080, 84231321290, 180920373632, 388598695916

Offset: 1

Eric W. Weisstein, Oct 27 2006

nonn

The antiprism graph is defined for n>=3; extended to n=1 using the closed form.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Max A. Alekseyev, Gérard P. Michon, Making Walks Count: From Silent Circles to Hamiltonian Cycles, arXiv:1602.01396 [math.CO], 2016-2017.
Mordecai J. Golin and Yiu Cho Leung, Unhooking Circulant Graphs: A Combinatorial Method for Counting Spanning Trees, Hamiltonian Cycles and other Parameters, Technical report HKUST-TCSC-2004-02. See also.
Eric Weisstein's World of Mathematics, Antiprism Graph
Eric Weisstein's World of Mathematics, Hamiltonian Cycle
Index entries for linear recurrences with constant coefficients, signature (3,-1,-2,0,1).

Cf. A124352.

I:=[6,18,32,58,112]; [n le 5 select I[n] else 3*Self(n-1) - Self(n-2) - 2*Self(n-3) + Self(n-5): n in [1..35]]; // Vincenzo Librandi, Feb 04 2016

Table[2 (2 n + RootSum[-1 - 2 # - #^2 + #^3 &, #^n &]), {n, 20}]
LinearRecurrence[{3, -1, -2, 0, 1}, {6, 18, 32, 58, 112}, 50] (* Vincenzo Librandi, Feb 04 2016 *)
Join[{6, 18}, Rest[Rest[Rest[CoefficientList[Series[-18*x^2 - 6*x - 6 + (4*x^2 + 4*x - 6)/(x^3 + 2*x^2 + x - 1) + 4/(x - 1)^2 + 4/(x - 1), {x, 0, 50}], x]]]]] (* G. C. Greubel, Apr 27 2017 *)

x='x+O('x^50); concat([6,18], Vec(-18*x^2-6*x-6+(4*x^2+4*x-6)/(x^3+2*x^2+x-1)+4/(x-1)^2+4/(x-1))) \\ G. C. Greubel, Apr 27 2017

a(n) = 3*a(n-1) - a(n-2) - 2*a(n-3) + a(n-5).
a(n) = 2*a(n-1) + a(n-2) - a(n-3) - a(n-4) - 12.
O.g.f.: -18*x^2-6*x-6+(4*x^2+4*x-6)/(x^3+2*x^2+x-1)+4/(x-1)^2+4/(x-1) . - R. J. Mathar, Feb 10 2008
a(n) = 2*(n + 3*A000930(2*n) - 2*A000930(2*n-1)) = A137725(2*n) = 2*A137726(2*n).

Formulas and further terms from Max Alekseyev, Feb 08 2008

Typo in formula corrected by Max Alekseyev, Nov 03 2010

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

Original entry on oeis.org

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

A287988 Number of (undirected) paths in the n-antiprism graph.

Original entry on oeis.org

2, 56, 396, 2040, 9130, 37944, 151172, 586608, 2235618, 8407640, 31292844, 115494312, 423283562, 1542120664, 5589611460, 20170172896, 72499928322, 259692909048, 927342338956, 3302291258200, 11730149911914, 41572470711288, 147031327493572, 519029653663056

Offset: 1

Eric W. Weisstein, Jun 03 2017

nonn

Sequence extrapolated to n=1 using recurrence. - Andrew Howroyd, Jun 05 2017

Andrew Howroyd, Table of n, a(n) for n = 1..200
Eric Weisstein's World of Mathematics, Antiprism Graph
Eric Weisstein's World of Mathematics, Graph Path
Index entries for linear recurrences with constant coefficients, signature (10, -37, 64, -58, 36, -26, 16, -5, 2, -1).

Cf. A077263, A124352, A124353, A287992.

Table[n RootSum[-1 - # - 3 #^2 + #^3 &, 23 #^n + 32 #^(n + 1) + 5 #^(n + 2) &]/44 - 7 n - 3 n^2 - 2 n^3, {n, 20}]
LinearRecurrence[{10, -37, 64, -58, 36, -26, 16, -5, 2, -1}, {2, 56, 396, 2040, 9130, 37944, 151172, 586608, 2235618, 8407640}, 20]
CoefficientList[Series[(2 (2 x^6 + 4 x^5 + x^4 + 24 x^3 - 4 x^2 + 20 x + 1) (1 - 2 x - x^2))/((1 - x)^4 (1 - 3 x - x^2 - x^3)^2), {x, 0, 20}], x]

Vec(2*(2*x^6+4*x^5+x^4+24*x^3-4*x^2+20*x+1)*(1-2*x-x^2)/((1-x)^4*(1-3*x-x^2-x^3)^2) + O(x^20)) \\ Andrew Howroyd, Jun 05 2017

From Andrew Howroyd, Jun 05 2017 (Start)
a(n) = 10*a(n-1)-37*a(n-2)+64*a(n-3) -58*a(n-4)+36*a(n-5)-26*a(n-6) +16*a(n-7)-5*a(n-8) +2*a(n-9)-a(n-10) for n>10.
G.f.: 2*x*(2*x^6+4*x^5+x^4+24*x^3-4*x^2+20*x+1) * (1-2*x-x^2) / ((1-x)^4 * (1-3*x-x^2-x^3)^2).
(End)

a(1)-a(2) and a(14)-a(24) from Andrew Howroyd, Jun 05 2017

A124354 Orders of the automorphisms groups of the n-antiprism graph.

Original entry on oeis.org

48, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, 156, 160, 164, 168, 172, 176, 180, 184, 188, 192, 196, 200, 204, 208, 212, 216, 220, 224, 228, 232, 236

Offset: 3

Eric W. Weisstein, Oct 27 2006

nonn

Eric Weisstein's World of Mathematics, Graph Automorphism
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A124352.

Join[{48},4Range[4,60]] (* Harvey P. Dale, Aug 10 2013 *)

a(3) = 48, otherwise a(n) = 4n.
From Chai Wah Wu, Apr 14 2024: (Start)
a(n) = 2*a(n-1) - a(n-2) for n > 5.
G.f.: x^3*(36*x^2 - 80*x + 48)/(x - 1)^2. (End)

A307933 Number of (undirected) Hamiltonian paths in the n-antiprism graph.

Original entry on oeis.org

120, 408, 1200, 3240, 8330, 20720, 50418, 120760, 285846, 670416, 1560728, 3611020, 8311110, 19042656, 43459344, 98838684, 224091320, 506660240, 1142669766, 2571214756, 5773744326, 12940614624, 28953267050, 64676245192, 144261049680, 321334401528, 714843635370, 1588357198980

Offset: 3

Eric W. Weisstein, May 06 2019

nonn

Eric Weisstein's World of Mathematics, Antiprism Graph
Eric Weisstein's World of Mathematics, Hamiltonian Path

Cf. A124352.

a(n) = A124352(n)/2.
Conjectures from Colin Barker, Mar 29 2020: (Start)
G.f.: 2*x^3*(60 - 96*x - 60*x^2 + 84*x^3 + 61*x^4 - 73*x^5 - 41*x^6 + 15*x^7 + 14*x^8) / ((1 - x)^3*(1 - x - 2*x^2 - x^3)^2).
a(n) = 5*a(n-1) - 6*a(n-2) - 4*a(n-3) + 7*a(n-4) + 5*a(n-5) - 5*a(n-6) - 3*a(n-7) + a(n-8) + a(n-9) for n>11.
(End)

a(30) corrected by Georg Fischer, Jan 25 2020

cp's OEIS Frontend

A124353 Number of (directed) Hamiltonian circuits on the n-antiprism graph.

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

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A287988 Number of (undirected) paths in the n-antiprism graph.

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A124354 Orders of the automorphisms groups of the n-antiprism graph.

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A307933 Number of (undirected) Hamiltonian paths in the n-antiprism graph.

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