A292429 Number of matchings in the n-Apollonian network.

10, 99, 38613, 377686178656, 28685416388231593782505635840000

Offset: 1

Author

Eric W. Weisstein, Sep 30 2017

Keywords

• nonn

Comments

Term a(6) has 90 decimal digits and a(7) has 264 decimal digits. - Andrew Howroyd, May 30 2025

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..8
• Eric Weisstein's World of Mathematics, Apollonian Network.
• Eric Weisstein's World of Mathematics, Independent Edge Set.
• Eric Weisstein's World of Mathematics, Matching.

Crossrefs

Cf. A297558.

Programs

PARI

\\ here m0..m3 are 0..3 outside vertices matched internally.
step(S,x)={my([m0,m1,m2,m3]=S); [ m0^3 + 3*m0^2*m1, 2*m0^2*m1 + 4*m0*m1^2 + 2*m0^2*m2 + m0^3*x,  m0^2*m2 + 3*m0*m1^2 + 8*m0*m1*m2 + m0^2*m3 + 3*m1^3 + 4*m0^2*m1*x, 6*m0*m1*m2 + 2*m1^3 + 12*m1^2*m2 + 6*m0*m2^2 + 6*m0*m1*m3 + 3*(m0^2*m2 + 3*m0*m1^2)*x ]}
a(n, x=1)={my(S=[1,0,0,0]); for(i=1, n, S=step(S,x)); sum(k=0, 3, binomial(3, k) * S[1+k]) + 3*(S[1]+S[2])*x} \\ Andrew Howroyd, May 30 2025

Extensions

a(5) onwards from Andrew Howroyd, May 30 2025