A302718 -id:A302718 - cp's OEIS frontend

A292002 Number of (undirected) paths in the n-Apollonian network.

Original entry on oeis.org

30, 1017, 1417992, 1475602431690, 887899295279284798114284, 580352523353299355481324176693737926736684735143

Offset: 1

Eric W. Weisstein, Sep 07 2017

nonn

a(7) has 96 decimal digits and a(8) has 191 decimal digits. - Andrew Howroyd, Jun 09 2025

Andrew Howroyd, Table of n, a(n) for n = 1..9
Eric Weisstein's World of Mathematics, Apollonian Network.
Eric Weisstein's World of Mathematics, Graph Path.

Cf. A302718, A307549.

a(4) onwards from Andrew Howroyd, Jun 09 2025

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

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Mar 25 2018

nonn

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)

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

Cf. A292002, A302718, A307549.

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

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

cp's OEIS Frontend

A292002 Number of (undirected) paths in the n-Apollonian network.

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