%I #38 Aug 06 2024 09:57:49

%S 6,18,32,58,112,220,450,938,1982,4220,9022,19332,41472,89022,191150,

%T 410506,881656,1893634,4067256,8735972,18763898,40302866,86566390,

%U 185935764,399371142,857808780,1842486536,3957474934,8500256470,18257692546,39215680080,84231321290,180920373632,388598695916

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

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

%H G. C. Greubel, <a href="/A124353/b124353.txt">Table of n, a(n) for n = 1..1000</a>

%H Max A. Alekseyev, Gérard P. Michon, <a href="http://arxiv.org/abs/1602.01396">Making Walks Count: From Silent Circles to Hamiltonian Cycles</a>, arXiv:1602.01396 [math.CO], 2016-2017.

%H Mordecai J. Golin and Yiu Cho Leung, <a href="http://www.cse.ust.hk/tcsc/RR/2004-02.ps.gz">Unhooking Circulant Graphs: A Combinatorial Method for Counting Spanning Trees, Hamiltonian Cycles and other Parameters</a>, Technical report HKUST-TCSC-2004-02. See <a href="https://citeseerx.ist.psu.edu/pdf/c743d0c638822fc3ea42610ae26661ae95d21b80">also</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AntiprismGraph.html">Antiprism Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HamiltonianCycle.html">Hamiltonian Cycle</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-2,0,1).

%F a(n) = 3*a(n-1) - a(n-2) - 2*a(n-3) + a(n-5).

%F a(n) = 2*a(n-1) + a(n-2) - a(n-3) - a(n-4) - 12.

%F 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

%F a(n) = 2*(n + 3*A000930(2*n) - 2*A000930(2*n-1)) = A137725(2*n) = 2*A137726(2*n).

%t Table[2 (2 n + RootSum[-1 - 2 # - #^2 + #^3 &, #^n &]), {n, 20}]

%t LinearRecurrence[{3, -1, -2, 0, 1}, {6, 18, 32, 58, 112}, 50] (* _Vincenzo Librandi_, Feb 04 2016 *)

%t 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 *)

%o (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

%o (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

%Y Cf. A124352.

%K nonn

%O 1,1

%A _Eric W. Weisstein_, Oct 27 2006

%E Formulas and further terms from _Max Alekseyev_, Feb 08 2008

%E Typo in formula corrected by _Max Alekseyev_, Nov 03 2010