A006396
Number of sensed planar maps with n edges and without faces or vertices of degree 1.
Original entry on oeis.org
1, 0, 1, 2, 4, 10, 36, 132, 616, 3060, 16207, 88990, 503816
Offset: 0
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A006400
Number of sensed simple planar maps with n edges and without vertices of degree 1.
Original entry on oeis.org
1, 0, 0, 1, 1, 2, 5, 11, 33, 117, 431, 1755, 7485
Offset: 0
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A054935
Number of planar maps with n edges up to orientation-preserving duality.
Original entry on oeis.org
1, 3, 7, 33, 156, 1070, 7515, 59151, 483925, 4136964, 36416865, 329048627, 3037029030, 28553451498, 272766018806, 2642420298576, 25916954091582, 257009789443925, 2573962338306141, 26008719387850068, 264933535266372732
Offset: 1
Average of
A006384 and
A006849, the latter interspersed with 0's (cf. formula).
-
a6384[0] = 1; a6384[n_] := (1/(2n))*(2*(3^n/((n + 1)*(n + 2)))*Binomial[2 n, n] + Sum[ EulerPhi[n/k]*3^k*Binomial[ 2k, k], {k, Most[ Divisors[ n]]}]) + q[n];
q[n_?OddQ] := 2*(3^((n - 1)/2)/(n + 1))*Binomial[ n - 1, (n - 1)/2];
q[n_?EvenQ] := 2*(n-1)*(3^((n-2)/2)/(n*(n+2)))*Binomial[ n - 2, (n - 2)/2];
a6849[n_] := 3^n*CatalanNumber[n]/2 + If[OddQ[n], 3^((n - 1)/2)* CatalanNumber[(n - 1)/2]/2, 0];
a[n_] := If[OddQ[n], a6384[n]/2, (a6384[n] + a6849[n/2])/2];
Array[a, 21] (* Jean-François Alcover, Aug 30 2019 *)
-
F(n) = { 3^n * binomial(2*n,n); }
S(n) = { my(acc = 0);
fordiv(n, d, if(d != n, acc += eulerphi(n/d) * F(d)));
return(acc); }
Q(n) = { if (n%2, 2 * F((n-1)/2) / (n+1),
2 * F((n-2)/2) * (n-1)/(n*(n+2))); }
A006384(n) = { if (n < 0, return(0)); if (n == 0, return(1));
(2*F(n)/((n+1)*(n+2)) + S(n)) / (2*n) + Q(n); }
G(n) = { 3^n * binomial(2*n,n) / (n + 1); }
A006849(n) = { if (n <= 0, return(0));
if (n%2, (G(n) + G((n-1)/2)) / 2, G(n)/2); }
a(n) = { if (n <= 0, return(0));
if (n%2, A006384(n)/2, (A006384(n) + A006849(n/2))/2) };
apply(n->a(n), vector(33, i, i)) \\ Gheorghe Coserea, Aug 20 2015
A103937
Number of unrooted n-edge maps in the plane (planar map with a distinguished outside face).
Original entry on oeis.org
1, 2, 6, 26, 150, 1032, 8074, 67086, 586752, 5317226, 49592424, 473357994, 4606116310, 45554761836, 456848968518, 4637014782748, 47563495004742, 492422043299964, 5140194991046122, 54053208147441474, 572191817441284272, 6093471300213162072, 65245904156725935906
Offset: 0
- V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 2005-01, Montreal, Canada, 2005.
-
a[n_] := (1/(2n)) (3^n Binomial[2n, n]/(n+1) + Sum[Boole[0Jean-François Alcover, Aug 28 2019 *)
-
a(n) = {if(n==0, 1, (3^n*binomial(2*n,n)/(n+1) + sumdiv(n, k, if(kAndrew Howroyd, Jan 22 2025
A006398
Number of sensed planar maps with n edges and without loops or isthmuses.
Original entry on oeis.org
1, 0, 1, 2, 4, 8, 29, 92, 403, 1846, 9239, 48028, 259567
Offset: 0
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A006404
Number of sensed 2-connected maps with n edges and without faces of degree 2.
Original entry on oeis.org
1, 1, 2, 5, 10, 29, 96, 339, 1320, 5473
Offset: 3
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A054937
Number of chiral pairs of planar maps with n edges.
Original entry on oeis.org
0, 0, 0, 0, 5, 64, 655, 5858, 51369, 448982, 3967466, 35603366, 324990677, 3016738988, 28449849867, 272233685444, 2639649712580, 25902435997188
Offset: 0
Comments