A192792 Molecular topological indices of the Apollonian graphs.

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

Offset: 1

Author

Eric W. Weisstein, Jul 10 2011

Keywords

• nonn

Links

• 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).

Crossrefs

Cf. A289022, A289722.

Programs

Mathematica

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 *)

PARI

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

Formula

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)

Extensions

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