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A289022 Wiener index of the n-Apollonian network. 5
6, 27, 204, 1941, 19572, 198567, 1999056, 19931337, 196939572, 1930784091, 18802964760, 182062831005, 1754100012108, 16826739416271, 160799296563312, 1531421717572401, 14540848734272388, 137690120683444995, 1300613432805623496, 12258142039717884549 (list; graph; refs; listen; history; text; internal format)
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

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 A047778

Adjacent sequences:  A289019 A289020 A289021 * A289023 A289024 A289025

KEYWORD

nonn

AUTHOR

Andrew Howroyd, Sep 02 2017

STATUS

approved

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Last modified September 27 15:29 EDT 2020. Contains 337383 sequences. (Running on oeis4.)