A298105 - OEIS

%I #13 Feb 16 2025 08:33:53

%S 15,96,30208,1562294353920,458842788759956138092569943104028672

%N Number of connected dominating sets in the n-Apollonian network.

%C Term a(6) has 107 decimal digits and a(7) has 320 decimal digits. - _Andrew Howroyd_, Jan 16 2018

%H Andrew Howroyd, <a href="/A298105/b298105.txt">Table of n, a(n) for n = 1..7</a>

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

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ConnectedDominatingSet.html">Connected Dominating Set</a>

%t {1, 3, 3, 1} . # & /@ NestList[Function[{t0, t1, t2, t3}, {t1^3, t1 t2^2, t1^2 t2 + t2^2 t3, t2^3 + t3^3}] @@ # &, {1, 2, 2, 2}, 4] (* _Eric W. Weisstein_, Jan 17 2018 *)

%o (PARI) \\ here t0..t3 are for 0..3 outside vertices included in set.

%o D(t0,t1,t2,t3)={[t1^3, t1*t2^2, t1^2*t2 + t2^2*t3, t2^3 + t3^3]}

%o a(n)={my(v=[1,2,2,2]); for(i=2, n, v=D(v[1], v[2], v[3], v[4])); v[1]+3*v[2]+3*v[3]+v[4]}

%Y Cf. A291773, A298123.

%K nonn,changed

%O 1,1

%A _Eric W. Weisstein_, Jan 12 2018

%E Terms a(4) and beyond from _Andrew Howroyd_, Jan 16 2018