A192792 - OEIS

%I #21 Sep 09 2017 10:38:27

%S 72,360,2556,22572,219636,2204244,22197420,222257988,2207645892,

%T 21754722852,212845625820,2069408197476,20010127994676,

%U 192565336573476,1845376043710284,17619057807964452,167667905660138532,1590879916369856484,15054743317985652924

%N Molecular topological indices of the Apollonian graphs.

%H Andrew Howroyd, <a href="/A192792/b192792.txt">Table of n, a(n) for n = 1..200</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ApollonianNetwork.html">Apollonian Network</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MolecularTopologicalIndex.html">Molecular Topological Index</a>

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (29, -320, 1676, -4109, 2537, 9182, -26346, 43920, -49896, 23328).

%F From _Andrew Howroyd_, Sep 03 2017: (Start)

%F 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.

%F 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)).

%F (End)

%t 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 *)

%t 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 *)

%t 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 *)

%o (PARI)

%o 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]}

%o Fin(peq, p1, p2, t, x) = {(t+1)*(peq + p1 + 2*p2 + x*(t + 3 + 2*p1 + p2))}

%o a(n) = { my(v=[18*x,x,0,0,3*x,0,0,2,x]);

%o for(i=2, n, v=Rec(v[1],v[2],v[3],v[4],v[5],v[6],v[7],v[8],x));

%o subst(deriv(v[1] + 3*Fin(v[2],v[3],v[4],v[8],x)), x, 1);

%o } \\ _Andrew Howroyd_, Sep 03 2017

%Y Cf. A289022, A289722.

%K nonn

%O 1,1

%A _Eric W. Weisstein_, Jul 10 2011

%E Terms a(7) and beyond from _Andrew Howroyd_, Sep 03 2017