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A124353 Number of (directed) Hamiltonian circuits on the n-antiprism graph. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

Max A. Alekseyev, GĂ©rard P. Michon, Making Walks Count: From Silent Circles to Hamiltonian Cycles, arXiv:1602.01396 [math.CO], 2016.

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.

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).

FORMULA

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).

MATHEMATICA

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 *)

PROG

(MAGMA) 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

(PARI) 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

CROSSREFS

Cf. A124352.

Sequence in context: A096286 A256256 A280802 * A232336 A153126 A110671

Adjacent sequences:  A124350 A124351 A124352 * A124354 A124355 A124356

KEYWORD

nonn

AUTHOR

Eric W. Weisstein, Oct 27 2006

EXTENSIONS

Formulas and further terms from Max Alekseyev, Feb 08 2008

Typo in formula corrected by Max Alekseyev, Nov 03 2010

STATUS

approved

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Last modified April 21 10:30 EDT 2019. Contains 322328 sequences. (Running on oeis4.)