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

1,1

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).

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)

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 *)

Prog

(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

Crossrefs

Cf. A289022, A289722.

Sequence in context: A223472 A090788 A334902 * A303621 A084479 A084478

Adjacent sequences: A192789 A192790 A192791 * A192793 A192794 A192795

Keyword

nonn

Author

Eric W. Weisstein, Jul 10 2011

Extensions

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

Status

approved