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 A194952 Number of Hamiltonian cycles in C_3 X C_n. 4
 48, 126, 390, 1014, 2982, 8094, 23646, 66726, 196086, 568302, 1682382, 4954998, 14750310, 43833150, 130942398, 390959430, 1170256854, 3502513038, 10495480494, 31450265622, 94296270918, 282731526366 (list; graph; refs; listen; history; text; internal format)
 OFFSET 3,1 COMMENTS All terms of this sequence are divisible by 6 (which follows from the g.f.). LINKS Vincenzo Librandi, Table of n, a(n) for n = 3..2000 Artem M. Karavaev, Hamilton Cycles page Eric Weisstein's World of Mathematics, Hamiltonian Cycle Eric Weisstein's World of Mathematics, Torus Grid Graph Index entries for linear recurrences with constant coefficients, signature (5,-1,-25,26,20,-24). FORMULA a(n) = 3^n + 3/4*n*2^n + (2^n-(-2)^n)/2 + (-1)^n - 4, n>=3. a(n) = 5*a(n-1)-a(n-2)-25*a(n-3)+26*a(n-4)+20*a(n-5)-24*a(n-6), for n>=9, with a(3)=48, a(4)=126, a(5)=390, a(6)=1014, a(7)=2982, a(8)=8094. G.f.: -6*x^3*(-8+19*x+32*x^2-65*x^3-34*x^4+48*x^5) / ( (x-1)*(3*x-1)*(2*x+1)*(1+x)*(-1+2*x)^2 ). - R. J. Mathar, Sep 18 2011 MAPLE C3xCn := n->3^n+3/4*n*2^n+(2^n-(-2)^n)/2+(-1)^n-4:seq(C3xCn(n), n=3..16); PROG (MAGMA) [3^n + 3/4*n*2^n + (2^n-(-2)^n)/2 + (-1)^n - 4: n in [3..40]]; // Vincenzo Librandi, Sep 19 2011 (Python) # Using graphillion from graphillion import GraphSet def make_CnXCk(n, k):     grids = []     for i in range(1, k + 1):         for j in range(1, n):             grids.append((i + (j - 1) * k, i + j * k))         grids.append((i + (n - 1) * k, i))     for i in range(1, k * n, k):         for j in range(1, k):             grids.append((i + j - 1, i + j))         grids.append((i + k - 1, i))     return grids def A194952(n):     universe = make_CnXCk(n, 3)     GraphSet.set_universe(universe)     cycles = GraphSet.cycles(is_hamilton=True)     return cycles.len() print([A194952(n) for n in range(3, 30)])  # Seiichi Manyama, Nov 22 2020 CROSSREFS Row 3 of A270273. Sequence in context: A260759 A211726 A232938 * A260362 A114444 A044299 Adjacent sequences:  A194949 A194950 A194951 * A194953 A194954 A194955 KEYWORD nonn,easy AUTHOR Artem M. Karavaev, Sep 06 2011 EXTENSIONS More terms from Alexander R. Povolotsky, Sep 07 2011 STATUS approved

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Last modified January 18 00:43 EST 2022. Contains 350410 sequences. (Running on oeis4.)