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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.
5
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
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
KEYWORD
nonn,tabf
AUTHOR
Andrew Howroyd, Sep 02 2017
STATUS
approved