A002712
Number of unrooted triangulations of a disk that have reflection symmetry with n interior nodes and 3 nodes on the boundary.
Original entry on oeis.org
1, 1, 1, 3, 8, 23, 68, 215, 680, 2226, 7327, 24607, 83060, 284046, 975950, 3383343, 11778308, 41269252, 145131502, 512881550, 1818259952, 6470758289, 23091680690, 82659905947, 296605398856, 1067012168350, 3846553544904, 13896522968160, 50296815014780, 182378110257354, 662384549806938
Offset: 0
- 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).
- Andrew Howroyd, Table of n, a(n) for n = 0..500
- William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.
- W. G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768. [Annotated scanned copy]
- C. F. Earl and L. J. March, Architectural applications of graph theory, pp. 327-355 of R. J. Wilson and L. W. Beineke, editors, Applications of Graph Theory. Academic Press, NY, 1979. (Annotated scanned copy)
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Dc := proc(n,m) 2*(2*m+3)!*(4*n+2*m+1)!/m!/(m+2)!/n!/(3*n+2*m+3)! ; end:
A000260 := proc(n) Dc(n,0) ; end:
Dx2 := proc(nmax) add( A000260(n)*x^(2*n),n=0..nmax) ; end:
o := 20: Order := 2*o-1 : j := solve( J0=1+x*J0+x^2*J0*(1+x*J0/2)*series(J0^2-Dx2(o),x=0,2*o-1),J0) ;
for n from 0 to 2*o-2 do printf("%d,",coeftayl(j,x=0,n)) ; od: # R. J. Mathar, Oct 29 2008
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seq[m_] := Module[{q}, q = Sum[x^(2n) Binomial[4n+2, n+1]/ ((2n+1)(3n+2)), {n, 0, Quotient[m, 2]}]; p = 1+O[x]; Do[p = 1 + x*p + x^2*p*(1+x*p/2)(p^2-q), {n, 1, m}]; CoefficientList[p, x]];
seq[30] (* Jean-François Alcover, Apr 25 2023, after Andrew Howroyd *)
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seq(n)={my(q=sum(n=0, n\2, x^(2*n)*binomial(4*n+2, n+1)/((2*n+1)*(3*n+2))), p=1+O(x)); for(n=1, n, p = 1 + x*p + x^2*p*(1 + x*p/2)*(p^2 - q)); Vec(p)} \\ Andrew Howroyd, Feb 24 2021
Name clarified and terms a(27) and beyond from
Andrew Howroyd, Feb 24 2021
A005498
Triangulations of the disk G_{2,n}.
Original entry on oeis.org
1, 6, 21, 88, 330, 1302, 5005, 19504, 75582, 294140, 1144066, 4458192, 17383860, 67866918, 265182525, 1037169760, 4059928950, 15905412468, 62359143990, 244662838160, 960566918220, 3773656396796, 14833897694226, 58343359313568, 229591913401900, 903936171565752, 3560597348629860
Offset: 0
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A005499
Triangulations of the disk G_{3,n}.
Original entry on oeis.org
5, 26, 119, 538, 2310, 9882, 41715, 175088, 730626, 3037510, 12584726, 52003792, 214401024, 882233898, 3624161175, 14865947668, 60898934250, 249184153548, 1018532686314, 4159265561360, 16970015555220, 69183689403686, 281844056190294, 1147419353238816, 4668368905854840, 18982659409726792
Offset: 0
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A253882
Number of 3-connected planar triangulations of the sphere with n vertices up to orientation preserving isomorphisms.
Original entry on oeis.org
1, 1, 2, 6, 17, 73, 389, 2274, 14502, 97033, 672781, 4792530, 34911786, 259106122, 1954315346, 14949368524, 115784496932, 906736988527, 7171613842488, 57231089062625, 460428456484557, 3731572377382341, 30447133566946517, 249968326771680542, 2063931874299323140
Offset: 4
- Andrew Howroyd, Table of n, a(n) for n = 4..500
- CombOS - Combinatorial Object Server, generate planar graphs
- Pascal Honvault, Equivalent classes of degree sequences for triangulated polyhedra and their convex realization, Contributions to Disc. Math. (2021) Vol. 16, No. 1, 128-137.
- Pascal Honvault, Local geometry of polyhedra, hal-03744217 [math], 2022.
- The House of Graphs, Planar graphs
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a(n)={if(n<3, 0, (2*binomial(4*(n-3)+1, n-3)/((n-2)*(3*n-7))
+ 3*sumdiv(n-2, d, if(d>=2, my(s=(n-2)/d); eulerphi(d)*binomial(4*s,s))/4)
+ if(n%2==1, my(s=(n-3)/2); 3*binomial(4*s,s)*(2*s+1)/(3*s+1))
+ if(n%3==1, my(s=(n-4)/3); 8*binomial(4*s,s)*(4*s+1)/(3*s+1))
+ if(n%3==0, my(s=(n-3)/3); 2*binomial(4*s,s)) )/(6*(n-2)))} \\ Andrew Howroyd, Mar 02 2021
Name clarified and terms a(24) and beyond from
Andrew Howroyd, Mar 02 2021
Original entry on oeis.org
1, 2, 5, 16, 48, 164, 599, 1952
Offset: 0
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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