A000264 Number of 3-edge-connected rooted cubic maps with 2n nodes and a distinguished Hamiltonian cycle. (Formerly M2974 N1203)

1, 1, 3, 14, 80, 518, 3647, 27274, 213480, 1731652, 14455408, 123552488, 1077096124, 9548805240, 85884971043, 782242251522, 7203683481720, 66989439309452, 628399635777936, 5940930064989720, 56562734108608536

Offset

1,3

References

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..200

L. B. Richmond, On Hamiltonian polygons, J. Combinatorial Theory Ser. B 21 (1976), no. 1, 81--87. MR0432491 (55 #5479) [See v_n].

W. T. Tutte, A census of Hamiltonian polygons, Canad. J. Math., 14 (1962), 402-417.

Formula

Let b(n)=(2n)!*(2n+2)!/(2*n!*(n+1)!^2*(n+2)!). Let B(x) be the generating function producing b(n), and A(x) be the generating function producing a(n). Then these sequences satisfy the functional equation B(x)=A(x(1+2*B(x))^2). - Sean A. Irvine, Apr 05 2010

Mathematica

max = 21; b[n_] := (2n)!*(2n + 2)!/(2*n!*(n + 1)!^2*(n + 2)!); b[0] = 0; bf[x_] := Sum[b[n]*x^n, {n, 0, max}]; Clear[a]; a[0] = 0; a[1] = a[2] = 1; af[x_] := Sum[a[n]*x^n, {n, 0, max}]; se = Series[bf[x] - af[x*(1 + 2*bf[x])^2], {x, 0, max}] // Normal; Table[a[n], {n, 1, max}] /. SolveAlways[se == 0, x] // First (* Jean-François Alcover, Jan 31 2013, after Sean A. Irvine *)

Crossrefs

Cf. A000309, A000356, A004304.

Sequence in context: A361770 A107596 A212391 * A009053 A377662 A202474

Adjacent sequences: A000261 A000262 A000263 * A000265 A000266 A000267

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

Better definition from Michael Albert, Oct 24 2008

More terms from Sean A. Irvine, Apr 05 2010

Status

approved