A058859 -id:A058859 - cp's OEIS frontend

A058860 Number of 2-connected rooted cubic planar maps with n faces.

1, 3, 19, 128, 909, 6737, 51683, 407802, 3293497, 27122967, 227095683, 1928656876, 16582719509, 144125955717, 1264625068163, 11190598332502, 99776445196977, 895685185070155, 8090065969366259, 73480719648381240, 670821169614526749

Offset: 4

N. J. A. Sloane, Jan 06 2001; revised Feb 17 2006

nonn

			G.f. = x^4 + 3*x^5 + 19*x^6 + 128*x^7 + 909*x^8 + 6737*x^9 + 51683*x^10 + ... - _Michael Somos_, Jul 22 2018

Gheorghe Coserea, Table of n, a(n) for n = 4..306
Z. Gao and N. C. Wormald, Enumeration of rooted cubic planar maps
Z. Gao and N. C. Wormald, Enumeration of rooted cubic planar maps, Annals of Combinatorics, 6 (2002), no. 3-4, 313-325.

Cf. A000260, A058859, A058861.

eq:=16*x^2*f^3+(8*x^4+24*x^3+72*x^2+8*x)*f^2+(x^6+6*x^5-5*x^4-40*x^3+3*x^2-14*x+1)*f-x^4-3*x^3+13*x^2-x: f:=sum(A[j]*x^j,j=1..35): for n from 1 to 35 do A[n]:=solve(coeff(expand(eq),x^n)=0) od: C2:=x^2*(f-x)*(1-2*x)/(1+x): C2ser:=series(C2,x=0,30): seq(coeff(C2ser,x^n),n=4..26); # Emeric Deutsch, Nov 30 2005

F = x^2*(z - x)*(1 - 2*x)/(1 + x);
G = 16*x^4*z^3 + x*(8*x^4 + 24*x^3 + 72*x^2 + 8*x)*z^2 + (x^6 + 6*x^5 -5*x^4 -40*x^3 + 3*x^2 - 14*x + 1)*z - x^3 - 3*x^2 + 13*x - 1;
Z(N) = {
  my(z0 = 1 + O('x^N), z1=0, n=1);
  while (n++,
    z1 = z0 - subst(G, 'z, z0)/subst(deriv(G, 'z), 'z, z0);
    if (z1 == z0, break()); z0 = z1); z0;
};
seq(N) = Vec(subst(F, 'z, 'x*Z(N+1)));
seq(21)
\\ test: y=Ser(seq(303),'x)*x^4; 0 == 16*y^3 - 8*x*(2*x - 1)*(x^2 + 8*x + 1)*y^2 + x^2*(2*x - 1)^2*(x^4 + 20*x^3 + 50*x^2 - 16*x + 1)*y - x^6*(2*x - 1)^3*(x^2 + 11*x - 1)
\\ Gheorghe Coserea, Jul 14 2018

G.f.: x^2*(f-x)*(1-2*x)/(1+x), where f is defined by 16*x^2*f^3 + (8*x^4+24*x^3+72*x^2+8*x)*f^2 + (x^6+6*x^5-5*x^4-40*x^3+3*x^2-14*x+1)*f - x^4-3*x^3+13*x^2-x=0. - Emeric Deutsch, Nov 30 2005
From Gheorghe Coserea, Jul 14 2018: (Start)
G.f. y=A(x) satisfies:
0 = 16*y^3 - 8*x*(2*x - 1)*(x^2 + 8*x + 1)*y^2 + x^2*(2*x - 1)^2*(x^4 + 20*x^3 + 50*x^2 - 16*x + 1)*y - x^6*(2*x - 1)^3*(x^2 + 11*x - 1).
0 = x^3*(2*x - 1)^3*(x - 2)*(4*x - 5)*(2*x^2 + 10*x - 1)*y''' - x^2*(2*x - 1)^2*(96*x^5 + 188*x^4 - 1570*x^3 + 1791*x^2 - 481*x + 35)*y'' + 12*x*(2*x - 1)*(48*x^6 + 104*x^5 - 898*x^4 + 1186*x^3 - 514*x^2 + 95*x - 5)*y' - 6*(256*x^7 + 608*x^6 - 5456*x^5 + 8292*x^4 - 4962*x^3 + 1525*x^2 - 220*x + 10)*y.
(End)

More terms from Emeric Deutsch, Nov 30 2005

A058861 Number of 3-connected rooted cubic planar maps with n faces and girth at least 4.

Original entry on oeis.org

0, 0, 1, 3, 12, 59, 313, 1713, 9559, 54189, 311460, 1812281, 10661303, 63336873, 379601353, 2293205687, 13953099573, 85451824382, 526431271347, 3260689089300, 20296848348929, 126918850161182, 796981464813540

Offset: 4

N. J. A. Sloane, Jan 06 2001; revised Feb 17 2006

nonn

Number of 3-connected triangle-free rooted cubic maps with n faces.

Gheorghe Coserea, Table of n, a(n) for n = 4..308
Z. Gao and N. C. Wormald, Enumeration of rooted cubic planar maps
Z. Gao and N. C. Wormald, Enumeration of rooted cubic planar maps, Annals of Combinatorics, 6 (2002), no. 3-4, 313-325.

Cf. A000260, A058859, A058860.

eq:=(x^3-3*x^2+3*x-1)*g^4+(4*x^4-12*x^3+9*x^2+2*x-3)*g^3+(6*x^5-10*x^4-15*x^3+36*x^2-14*x-3)*g^2+(4*x^6+4*x^5-45*x^4+82*x^3-59*x^2+14*x-1)*g+x^7+5*x^6-8*x^5+x^4: g:=sum(A[j]*x^j,j=1..37): for n from 1 to 37 do A[n]:=solve(coeff(expand(eq),x^n)=0) od: C3:=x^2*(1-3*x)*g: C3ser:=series(C3,x=0,34): seq(coeff(C3ser,x^n),n=6..30); # Emeric Deutsch, Nov 30 2005

F = x^2*(1 - 3*x)*z;
G = x^12*(x - 1)^3*z^4 + x^8*(x - 1)^2*(2*x - 3)*(2*x + 1)*z^3 + x^4*(x - 1)*(6*x^4 - 4*x^3 - 19*x^2 + 17*x + 3)*z^2 + (4*x^6 + 4*x^5 - 45*x^4 + 82*x^3 - 59*x^2 + 14*x - 1)*z + (x^3 + 5*x^2 - 8*x + 1);
Z(N) = {
  my(z0 = 1 + O('x^N), z1=0, n=1);
  while (n++,
    z1 = z0 - subst(G, 'z, z0)/subst(deriv(G, 'z), 'z, z0);
    if (z1 == z0, break()); z0 = z1); z0;
};
seq(N) = concat([0, 0], Vec(subst(F, 'z, 'x^4*Z(N))));
seq(21)
\\ test: y = Ser(seq(303))*'x^4; 0 == (x - 1)^3*y^4 - x^2*(x - 1)^2*(2*x - 3)*(2*x + 1)*(3*x - 1)*y^3 + x^4*(x - 1)*(3*x - 1)^2*(6*x^4 - 4*x^3 - 19*x^2 + 17*x + 3)*y^2 - x^6*(3*x - 1)^3*(4*x^6 + 4*x^5 - 45*x^4 + 82*x^3 - 59*x^2 + 14*x - 1)*y + x^12*(3*x - 1)^4*(x^3 + 5*x^2 - 8*x + 1)
\\ Gheorghe Coserea, Jul 15 2018

G.f.: x^2*(1-3*x)*g, where g is defined by (x^3-3*x^2+3*x-1)*g^4 + (4*x^4-12*x^3+9*x^2+2*x-3)*g^3 + (6*x^5-10*x^4-15*x^3+36*x^2-14*x-3)*g^2 + (4*x^6+4*x^5-45*x^4+82*x^3-59*x^2+14*x-1)*g + x^7+5*x^6-8*x^5+x^4=0. - Emeric Deutsch, Nov 30 2005
From Gheorghe Coserea, Jul 15 2018: (Start)
G.f. y=A(x) satisfies:
0 = (x - 1)^3*y^4 - x^2*(x - 1)^2*(2*x - 3)*(2*x + 1)*(3*x - 1)*y^3 + x^4*(x - 1)*(3*x - 1)^2*(6*x^4 - 4*x^3 - 19*x^2 + 17*x + 3)*y^2 - x^6*(3*x - 1)^3*(4*x^6 + 4*x^5 - 45*x^4 + 82*x^3 - 59*x^2 + 14*x - 1)*y + x^12*(3*x - 1)^4*(x^3 + 5*x^2 - 8*x + 1).
0 = x^4*(x - 1)^3*(3*x - 1)^4*(256*x^3 - 512*x^2 + 256*x - 27)*(660*x^5 - 2668*x^4 + 4177*x^3 - 3252*x^2 + 1305*x - 220)*y'''' - 4*x^3*(x - 1)^2*(3*x - 1)^3*(696960*x^10 - 5684352*x^9 + 19870624*x^8 - 39218578*x^7 + 48478923*x^6 - 39311914*x^5 + 21210466*x^4 - 7501069*x^3 + 1650104*x^2 - 201370*x + 10395)*y''' + 4*x^2*(x - 1)*(3*x - 1)^2*(7579440*x^12 - 77500656*x^11 + 341548428*x^10 - 862746936*x^9 + 1396393806*x^8 - 1530275829*x^7 + 1167408906*x^6 - 625929723*x^5 + 234247228*x^4 - 59616890*x^3 + 9784582*x^2 - 931830*x + 38940)*y'' - 24*x*(3*x - 1)*(8523900*x^14 - 105292620*x^13 + 561229815*x^12 - 1731677190*x^11 + 3479254732*x^10 - 4837165728*x^9 + 4815815835*x^8 - 3498631418*x^7 + 1868972298*x^6 - 732379803*x^5 + 207693098*x^4 - 41430916*x^3 + 5510394*x^2 - 438095*x + 15730)*y' + 24*(27442800*x^15 - 372895380*x^14 + 2140330050*x^13 - 7047776880*x^12 + 15074631336*x^11 - 22357962673*x^10 + 23891962029*x^9 - 18825921582*x^8 + 11080006886*x^7 - 4892373579*x^6 + 1614037497*x^5 - 392156906*x^4 + 68130318*x^3 - 8004294*x^2 + 569210*x - 18480)*y.
(End)

More terms from Emeric Deutsch, Nov 30 2005

cp's OEIS Frontend

A058860 Number of 2-connected rooted cubic planar maps with n faces.

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