A289722 -id:A289722 - cp's OEIS frontend

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

Andrew Howroyd, Sep 02 2017

nonn

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).

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

(* 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 *)

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));

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)).

A192792 Molecular topological indices of the Apollonian graphs.

Original entry on oeis.org

72, 360, 2556, 22572, 219636, 2204244, 22197420, 222257988, 2207645892, 21754722852, 212845625820, 2069408197476, 20010127994676, 192565336573476, 1845376043710284, 17619057807964452, 167667905660138532, 1590879916369856484, 15054743317985652924

Offset: 1

Eric W. Weisstein, Jul 10 2011

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..200
Eric Weisstein's World of Mathematics, Apollonian Network
Eric Weisstein's World of Mathematics, Molecular Topological Index
Index entries for linear recurrences with constant coefficients, signature (29, -320, 1676, -4109, 2537, 9182, -26346, 43920, -49896, 23328).

Cf. A289022, A289722.

Table[(3 (3025 + 605 2^(2 + n) + 3 (-1)^n 2^(4 + n) + 605 2^(3 + 2 n) + 5 9^n (63 + 44 n) + 4 3^n (-277 + 55 n) + 35 2^(2 + n/2) Cos[n Pi/2] - 15 2^((7 + n)/2) Sin[n Pi/2]))/1210, {n, 20}]  (* Eric W. Weisstein, Sep 08 2017 *)
LinearRecurrence[{29, -320, 1676, -4109, 2537, 9182, -26346, 43920, -49896, 23328}, {72, 360, 2556, 22572, 219636, 2204244, 22197420, 222257988, 2207645892, 21754722852}, 20]  (* Eric W. Weisstein, Sep 08 2017 *)
CoefficientList[Series[36 (2 - 48 x + 421 x^2 - 1584 x^3 + 2096 x^4 + 1960 x^5 - 9573 x^6 + 17670 x^7 - 25056 x^8 + 15552 x^9)/((1 - x) (1 - 2 x) (1 - 3 x)^2 (1 - 4 x) (1 - 9 x)^2 (1 + 2 x) (1 + 2 x^2)), {x, 0, 50}], x]  (* Eric W. Weisstein, Sep 08 2017 *)

Rec(mti, peq, p1, p2, weq, w1, w2, t, x) = {[3*(mti + 2*weq*peq + 2*(2+7*x)*w1*p1 + 2*(7+2*x)*w2*p2 + (4+2*x)*(weq*p1+peq*w1) + 6*(weq*p2+peq*w2) + 2*(4+5*x)*(w1*p2+p1*w2) + x*(weq+3*w1+3*w2) + 3*t*(peq+p1+2*p2) + 3*t*x*(t+1+2*p1+p2)), x*(1+3*p1), 2*(p1+p2), peq+p2, x*(3*t+3*w1), 2*(w1+w2), weq+w2, 2*t]}
Fin(peq, p1, p2, t, x) = {(t+1)*(peq + p1 + 2*p2 + x*(t + 3 + 2*p1 + p2))}
a(n) = { my(v=[18*x,x,0,0,3*x,0,0,2,x]);
for(i=2, n, v=Rec(v[1],v[2],v[3],v[4],v[5],v[6],v[7],v[8],x));
subst(deriv(v[1] + 3*Fin(v[2],v[3],v[4],v[8],x)), x, 1);
} \\ Andrew Howroyd, Sep 03 2017

From Andrew Howroyd, Sep 03 2017: (Start)
a(n) = 29*a(n-1) - 320*a(n-2) + 1676*a(n-3) - 4109*a(n-4) + 2537*a(n-5) + 9182*a(n-6) - 26346*a(n-7) + 43920*a(n-8) - 49896*a(n-9) + 23328*a(n-10) for n > 10.
G.f.: 36*x*(2 - 48*x + 421*x^2 - 1584*x^3 + 2096*x^4 + 1960*x^5 - 9573*x^6 + 17670*x^7 - 25056*x^8 + 15552*x^9)/((1 - x)*(1 - 2*x)*(1 - 3*x)^2*(1 - 4*x)*(1 - 9*x)^2*(1 + 2*x)*(1 + 2*x^2)).
(End)

Terms a(7) and beyond from Andrew Howroyd, Sep 03 2017

cp's OEIS Frontend

A289022 Wiener index of the n-Apollonian network.

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