A289722 Irregular triangle read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the n-Apollonian network.

6, 15, 6, 42, 72, 6, 123, 522, 258, 366, 2970, 3894, 396, 1095, 14838, 37332, 13680, 216, 3282, 68736, 278490, 224928, 24624, 9843, 303918, 1779678, 2517228, 754704, 22032, 29526, 1303938, 10269150, 22233096, 13114656, 1489104, 7776

Offset: 1

Author

Andrew Howroyd, Sep 02 2017

Keywords

• nonn
• tabf

Comments

Row n has k in the range 1..1+floor(2*n/3).

Table gives the coefficients of the Wiener polynomials. The degree of the polynomial corresponds to the diameter of the graph.

Examples

			Triangle begins:
6;
15, 6;
42, 72, 6;
123, 522, 258;
366, 2970, 3894, 396;
1095, 14838, 37332, 13680, 216;
3282, 68736, 278490, 224928, 24624;
9843, 303918, 1779678, 2517228, 754704, 22032;
29526, 1303938, 10269150, 22233096, 13114656, 1489104, 7776;
...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..883
• Eric Weisstein's World of Mathematics, Apollonian Network
• Eric Weisstein's World of Mathematics, Wiener Index

Crossrefs

Cf. A289022.

Programs

Mathematica

R[dp_, peq_, p1_, p2_] := {3*(dp - x + peq^2 + (2 + 7*x)*p1^2 + (7 + 2*x)*p2^2 + (4 + 2*x)*peq*p1 + 6*peq*p2 + 2*(4 + 5*x)*p1*p2 + x*(peq + 3*p1 + 3*p2)), x*(1 + 3*p1), 2*(p1 + p2), peq + p2};
A[n_] := (v = {6*x, x, 0, 0}; For[i = 2, i <= n, i++, v = R @@ v]; v[[1]]);
Table[CoefficientList[A[n], x] // Rest, {n, 1, 10}] // Flatten (* Jean-François Alcover, Dec 28 2017, after Andrew Howroyd *)

PARI

R(dp, peq,p1,p2,x) = {[3*(dp - x + peq^2 + (2+7*x)*p1^2 + (7+2*x)*p2^2 + (4+2*x)*peq*p1 + 6*peq*p2 + 2*(4+5*x)*p1*p2 + x*(peq+3*p1+3*p2)), x*(1+3*p1), 2*(p1+p2), peq+p2]}
A(n,x) = {my(v=[6*x,x,0,0,x]); for(i=2, n, v=R(v[1],v[2],v[3],v[4],x)); v[1]}
for (n=1,10,print(Vec(polrecip(A(n,x))),";" ))