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A124353
Number of (directed) Hamiltonian circuits on the n-antiprism graph.
7
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
1,1
COMMENTS
The antiprism graph is defined for n>=3; extended to n=1 using the closed form.
LINKS
Max A. Alekseyev, GĂ©rard P. Michon, Making Walks Count: From Silent Circles to Hamiltonian Cycles, arXiv:1602.01396 [math.CO], 2016-2017.
Eric Weisstein's World of Mathematics, Antiprism Graph
Eric Weisstein's World of Mathematics, Hamiltonian Cycle
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: A256256 A344596 A280802 * A232336 A153126 A110671
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