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A090372 Number of unrooted planar 3-constellations with n triangles. 3
1, 6, 22, 174, 1479, 16808, 201834, 2631594, 35965555, 512062566, 7528425420, 113708935808, 1756853846316, 27676951028496, 443411345677658, 7209139541742750, 118738765611199983, 1978360119497335826 (list; graph; refs; listen; history; text; internal format)



These are planar maps with bicolored faces having n black triangular faces and an arbitrary number of white faces of degrees multiple to 3. The vertices can be and are colored so that any black triangle is colored counterclockwise 1,2,3. Isomorphisms are required to respect the colorings. Also unrooted bi-Eulerian maps with bicolored both vertices and faces and with 2n edges; the maps are considered up to color-preserve isomorphism.


Table of n, a(n) for n=1..18.

M. Bousquet-Mélou and G. Schaeffer, Enumeration of planar constellations, Adv. in Appl. Math. v.24 (2000), 337-368.

V. A. Liskovets, Enumerative formulas for unrooted planar maps: a pattern, Electron. J. Combin., 11:1 (2004), R88.


with(numtheory): C_3 := proc(n) local s, d; if n=0 then RETURN(1) else s := -3^n*binomial(3*n, n); for d in divisors(n) do s := s+phi(n/d)*3^d*binomial(3*d, d) od; RETURN((4/(3*n))*(3^n*binomial(3*n, n)/((2*n+1)*(2*n+2))+s/2)); fi; end;


a[0] = 1; a[n_] := Module[{s, d}, s = -3^n Binomial[3n, n]; Do[s = s + EulerPhi[n/d] 3^d Binomial[3d, d], {d, Divisors[n]}]; (4/(3n)) (3^n Binomial[3n, n]/((2n+1)(2n+2)) + s/2)];

Array[a, 18] (* Jean-François Alcover, Jul 24 2018, from Maple *)


Cf. A069729, A090373.

Sequence in context: A075759 A000993 A028406 * A009366 A230964 A075811

Adjacent sequences:  A090369 A090370 A090371 * A090373 A090374 A090375




Valery A. Liskovets, Dec 01 2003



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Last modified November 14 12:32 EST 2019. Contains 329114 sequences. (Running on oeis4.)