login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A064521 Number of rooted 5-connected planar triangulations with 2n faces. 0
1, 0, 6, 13, 55, 189, 694, 2516, 9213, 33782, 124300, 458502, 1695469, 6284175, 23344173, 86904615, 324197100, 1211841846, 4538611107, 17029834923, 64014608376, 241046175666, 909171583214, 3434698413540, 12995770332449 (list; graph; refs; listen; history; text; internal format)
OFFSET
10,3
COMMENTS
No planar triangulation can be more than 5-connected. The 5-connected triangulations are historically important to the 4-color problem.
LINKS
Z. J. Gao, I. M. Wanless and N. C. Wormald, Counting 5-connected planar triangulations, J. Graph Theory, Vol. 38 (2001), pp. 18-35.
EXAMPLE
The smallest 5-connected planar triangulation is the icosahedron, which has 20 faces. Because of its symmetry it has a unique rooting, so a(10)=1. The triangulations counted by a(12) and a(13) are drawn in the paper cited above.
MAPLE
# G.f. for 5-connected planar triangulations: fiveconntri(m) returns the first m terms of a power series in w, in which the coefficient of w^n is the number of (rooted) 5-connected planar triangulations with 2n faces.
fiveconntri := proc(howmanyterms) local keepterms, T, iteration, sval, previous; keepterms := howmanyterms+1; T := -3*w^3/(1+w)+w-w^2+3*w^3-w^4+4*(s+1)^3*((3*s-1)*w+(3*s-2)*(s+1)^3)*w/((3*s+2+w-s^3)^3); iteration := s-(-w^2+2*(4*s^2+2*s+1)*(s+1)^2*w+s*(s+2)*(s+1)^4)/(8*w+2); sval := 0; previous := -1; while(sval<>previous) do previous := sval; sval := mtaylor(subs(s=sval, iteration), [w, s], keepterms); od: series(subs(s=sval, T), w, keepterms); end;
CROSSREFS
Sequence in context: A187985 A320043 A296619 * A367663 A330283 A361244
KEYWORD
nonn
AUTHOR
Ian Wanless, Oct 07 2001
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)