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

1,1

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.

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

Example

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;
...

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 *)

Prog

(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))), "; " ))

Crossrefs

Cf. A289022.

Sequence in context: A003566 A349083 A205149 * A351369 A070870 A202749

Adjacent sequences: A289719 A289720 A289721 * A289723 A289724 A289725

Keyword

nonn,tabf

Author

Andrew Howroyd, Sep 02 2017

Status

approved