|
|
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
|
|
|
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
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|