%N Number of rooted 5-connected planar triangulations with 2n faces.
%C No planar triangulation can be more than 5-connected. The 5-connected triangulations are historically important to the 4-color problem.
%H Z. J. Gao, I. M. Wanless and N. C. Wormald, <a href="https://doi.org/10.1002/jgt.1021">Counting 5-connected planar triangulations</a>, J. Graph Theory, Vol. 38 (2001), pp. 18-35.
%e 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.
%p # 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.
%p 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;
%A _Ian Wanless_, Oct 07 2001