A289022 Wiener index of the n-Apollonian network.

6, 27, 204, 1941, 19572, 198567, 1999056, 19931337, 196939572, 1930784091, 18802964760, 182062831005, 1754100012108, 16826739416271, 160799296563312, 1531421717572401, 14540848734272388, 137690120683444995, 1300613432805623496, 12258142039717884549

Offset

1,1

Links

Andrew Howroyd, Table of n, a(n) for n = 1..100

Eric Weisstein's World of Mathematics, Apollonian Network

Eric Weisstein's World of Mathematics, Wiener Index

Index entries for linear recurrences with constant coefficients, signature (23, -174, 448, -29, -1221, 2088, -4050, 2916).

Formula

a(n) = Sum_{k=1..1+floor(2*n/3)} k*A289722(n,k).

a(n) = 23*a(n-1) - 174*a(n-2) + 448*a(n-3) - 29*a(n-4) - 1221*a(n-5) + 2088*a(n-6) - 4050*a(n-7) + 2916*a(n-8).

G.f.: x*(6 - 111*x + 627*x^2 - 741*x^3 - 1497*x^4 + 2862*x^5 - 5670*x^6 + 8748*x^7)/((1 - x)*(1 - 3*x)^2*(1 - 9*x)^2*(1 + 2*x)*(1 + 2*x^2)).

Mathematica

(* Start from Eric W. Weisstein, Sep 07 2017 *)
Table[(6655 + 31 (-1)^n 2^(n + 2) + 5 3^(1 + 2 n) (24 + 11 n) + 3^(n + 1) (1197 + 55 n) + 5 2^(5 + n/2) Cos[n Pi/2] - 155 2^((3 + n)/2) Sin[n Pi/2])/3630, {n, 20}]
LinearRecurrence[{23, -174, 448, -29, -1221, 2088, -4050, 2916}, {6, 27, 204, 1941, 19572, 198567, 1999056, 19931337}, 20]
CoefficientList[Series[(6 - 111 x + 627 x^2 - 741 x^3 - 1497 x^4 + 2862 x^5 - 5670 x^6 + 8748 x^7)/((1 - x) (1 - 3 x)^2 (1 - 9 x)^2 (1 + 2 x) (1 + 2 x^2)), {x, 0, 20}], x]
(* End *)

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]}
Wiener(dp)=sum(i=1, poldegree(dp), i*polcoeff(dp, i));
a(n) = Wiener(A(n, x));

Crossrefs

Cf. A067771 (edges), A192792, A289521, A289722.

Sequence in context: A318565 A092854 A223557 * A060977 A267630 A351737

Adjacent sequences: A289019 A289020 A289021 * A289023 A289024 A289025

Keyword

nonn

Author

Andrew Howroyd, Sep 02 2017

Status

approved