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A301650 Number of longest cycles in the n-Apollonian network.

%I A301650 #10 Feb 16 2025 08:33:53
%S A301650 3,12,162,354294,1694577218886,38766491335360039793593446,
%T A301650 20288351481136358057581328834353447021191164711091366
%N A301650 Number of longest cycles in the n-Apollonian network.
%C A301650 From _Andrew Howroyd_, Sep 09 2019: (Start)
%C A301650 a(8) has 106 decimal digits and a(9) has 213 decimal digits.
%C A301650 The circumference or length of the longest cycle is given by 7*2^(n-2) for n > 1. For n = 1, the circumference is 4. (End)
%H A301650 Andrew Howroyd, <a href="/A301650/b301650.txt">Table of n, a(n) for n = 1..10</a>
%H A301650 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ApollonianNetwork.html">Apollonian Network</a>
%H A301650 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>
%o A301650 (PARI)
%o A301650 P(c,d,x)={[d^2 + 6*c*d + 2*d^3 + 2*x*(c + 3*d^2) + 2*x^2*d, c + d + 3*d^2 + 4*x*d + x^2]}
%o A301650 R(c,d,x)={4*d^3 + 9*c*d^2 + 3*d^2 + 6*c*d + 3*c^2 + 6*x*(2*d^3 + 3*d^2 + 4*c*d) + 3*x^2*(10*d^2 + 3*d + 3*c) + x^3*(18*d + 1) + 3*x^4}
%o A301650 a(n)={my(s=x^3, c=0, d=0); for(i=1, n, s = 3*s + R(c,d,x); [c,d]=P(c,d,x)); pollead(s)} \\ _Andrew Howroyd_, Sep 10 2019
%Y A301650 Cf. A292002, A302718, A307549.
%K A301650 nonn
%O A301650 1,1
%A A301650 _Eric W. Weisstein_, Mar 25 2018
%E A301650 a(5)-a(7) from _Andrew Howroyd_, Sep 09 2019