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A027610
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The number of Apollonian networks (planar 3-trees) with n+3 vertices.
(Formerly M2688)
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26
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1, 1, 1, 3, 7, 24, 93, 434, 2110, 11002, 58713, 321776, 1792133, 10131027, 57949430, 334970205, 1953890318, 11489753730, 68054102361, 405715557048, 2433003221232, 14668536954744, 88869466378593, 540834155878536, 3304961537938269, 20273202069859769
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OFFSET
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1,4
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COMMENTS
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Previous name was: Number of chordal planar triangulations; also number of planar triangulations with maximal number of triangles; also number of graphs obtained from the tetrahedron by repeatedly inserting vertices of degree 3 into a triangular face; also number of uniquely 4-colorable planar graphs; also number of simplicial 3-clusters with n cells; also Apollonian networks with n+3 vertices.
Also arises in enumeration of spectral isomers of alkane systems (see Cyvin et al.). - N. J. A. Sloane, Aug 15 2006
Chordal planar triangulations: take planar triangulations on n nodes, divide them into classes according to how many triangles they contain (all have 2n-4 triangular faces but may have additional triangles); count triangulations in class with most triangles.
If mirror images are not taken as equivalent, A007173 is obtained instead. - Brendan McKay, Mar 08 2014
Number of unoriented polyominoes composed of n tetrahedral cells of the hyperbolic regular tiling with Schläfli symbol {3,3,oo}. For unoriented polyominoes, chiral pairs are counted as one. - Robert A. Russell, Mar 20 2024
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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a(n) = C(3n,n)/(6*(2n+1)*(2n+2)) + ([0==n mod 2]*7*C(3n/2,n) + [1==n mod 2]*9*C((3n-1)/2,n)) / (12(n+1)) + [1==n mod 3]*C(n-1,(n-1)/3)/(2n+1) + [2==n mod 3]*C(n-1,(n-2)/3)/(2n+2) + [1==n mod 4]*C((3n-3)/4,(n-1)/2)/(2n+2) + [2==n mod 6]*C(n/2-1,(n-2)/3)/(2n+2).
a(n) = h(3,n) in Table 8 of Hering link.
G.f.: (-16 + 4*G(z) - 2*G(z)^2 + z*G(z)^4 + 14*G(z^2) + 9z*G(z^2)^2 + 8z*G(z^3) + 4z^2*G(z^3)^2 + 6z*G(z^4) + 4z^2*G(z^6))/24, where G(z) = 1 + z*G(z)^3 is the g.f. for A001764. (End)
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MAPLE
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A001764 := proc(m) RETURN((3*m)!/(m!*(2*m+1)!)); end; # Gives A001764(m)
A047749 := proc(m) local x; if m mod 2 = 1 then x := (m-1)/2; RETURN((3*x+1)!/((x+1)!*(2*x+1)!)); fi; RETURN(A001764(m/2)); end; # Gives A047749(m)
A027610 := proc(n) local N; N := 0; N := N + A001764(n)/(12*(n+1)); if n mod 2 = 0 then N := N + 5/24*A001764(n/2); fi; if (n-1) mod 3 = 0 then N := N + 1/3*A001764((n-1)/3); fi; if (n-1) mod 4 = 0 then N := N + 1/4*A001764((n-1)/4); fi;
if (n-2) mod 6 = 0 then N := N + 1/6*A001764((n-2)/6); fi; N := N + 3/8*A047749(n); if (2*n-1) mod 3 = 0 then N := N + 1/6*A047749((2*n-1)/3); fi; RETURN(N); end;
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MATHEMATICA
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Table[Binomial[3 n, 2 n]/(6 (2 n + 1) (2 n + 2)) + If[EvenQ[n], 7 Binomial[3 n/2, n]/(12 (n + 1)), 3 Binomial[3 n/2 - 1/2, n]/(4 (n + 1))] + Switch[Mod[n, 3], 1, Binomial[n - 1, 2 n/3 - 2/3]/(2 n/3 + 1/3), 2, Binomial[n - 1, 2 n/3 - 1/3]/(2 n/3 + 2/3), _, 0]/3 + If[1 == Mod[n, 4], Binomial[3 n/4 - 3/4, n/2 - 1/2]/(n/2 + 1/2), 0]/4 + If[2 == Mod[n, 6], Binomial[n/2 - 1, n/3 - 2/3]/(n/3 + 1/3), 0]/6, {n, 1, 30}] (* Robert A. Russell, Apr 11 2012 *)
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PROG
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(PARI) T(m)={if(m<0||denominator(m)!=1, 0, (3*m)!/(m!*(2*m+1)!))};
U(k)={if(k<0||denominator(k)!=1, 0, if(k%2, my(m=(k-1)/2); (3*m+1)!/((m+1)!*(2*m+1)!), T(k/2)))};
S(n)=T(n)/(12*(n+1))+5*T(n/2)/24+T((n-1)/3)/3+T((n-1)/4)/4+T((n-2)/6)/6+3*U(n)/8+U((2*n-1)/3)/6;
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CROSSREFS
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Sum of A047776, A047775, A047774, A047773, A047762, A047760, A047758, A047754, A047753, A047752, A047751, A047771, A047769, A047766 (twice), A047765, A047764.
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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Noted the name "Apollonian network" by Brendan McKay, Mar 08 2014
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STATUS
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approved
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