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

3, 12, 162, 354294, 1694577218886, 38766491335360039793593446, 20288351481136358057581328834353447021191164711091366

Offset: 1

Author

Eric W. Weisstein, Mar 25 2018

Keywords

• nonn

Comments

From Andrew Howroyd, Sep 09 2019: (Start)

a(8) has 106 decimal digits and a(9) has 213 decimal digits.

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)

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..10
• Eric Weisstein's World of Mathematics, Apollonian Network
• Eric Weisstein's World of Mathematics, Graph Cycle

Crossrefs

Cf. A292002, A302718, A307549.

Programs

PARI

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]}
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}
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

Extensions

a(5)-a(7) from Andrew Howroyd, Sep 09 2019