10,3

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.

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: A041489 A131188 A003757 this_sequence A111366 A119110 A041305

Adjacent sequences: A064518 A064519 A064520 this_sequence A064522 A064523 A064524

nonn

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