This site is supported by donations to The OEIS Foundation.



Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

(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)



No planar triangulation can be more than 5-connected. The 5-connected triangulations are historically important to the 4-color problem.


Z. J. Gao, I. M. Wanless and N. C. Wormald, Counting 5-connected planar triangulations, J. Graph Theory, Vol. 38 (2001), pp. 18-35.


Table of n, a(n) for n=10..34.


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.


# 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;


Sequence in context: A203977 A003757 A187985 * A262238 A111366 A177127

Adjacent sequences:  A064518 A064519 A064520 * A064522 A064523 A064524




Ian M. Wanless (wanless(AT)maths.ox.ac.uk), Oct 07 2001



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

License Agreements, Terms of Use, Privacy Policy .

Last modified December 8 12:55 EST 2016. Contains 278945 sequences.