A307457 Longest path length in the n-Apollonian network.

3, 6, 14, 30, 62, 126, 254, 510, 1022, 2046, 4094, 8190, 16382, 32766, 65534, 131070, 262142, 524286, 1048574, 2097150, 4194302, 8388606, 16777214, 33554430, 67108862, 134217726, 268435454, 536870910, 1073741822, 2147483646, 4294967294, 8589934590

Offset

1,1

Links

Paolo Xausa, Table of n, a(n) for n = 1..2500

Eric Weisstein's World of Mathematics, Apollonian Network.

Eric Weisstein's World of Mathematics, Longest Path.

Index entries for linear recurrences with constant coefficients, signature (3,-2).

Formula

a(n) = 2^(n+1) - 2 = A000918(n+1) for n > 1. - Andrew Howroyd, Jun 09 2025

From Elmo R. Oliveira, Nov 22 2025: (Start)

G.f.: x*(3 - 3*x + 2*x^2)/((2*x-1)*(x-1)).

E.g.f.: x + 2*exp(x)*(exp(x) - 1).

a(n) = 3*a(n-1) - 2*a(n-2) for n > 3. (End)

Mathematica

Join[{3}, 2^Range[3, 35] - 2] (* Paolo Xausa, Nov 22 2025 *)

Prog

(PARI) apply( {A307457(n)=2^(n+1)-1-(n>1)}, [1..33]) \\ M. F. Hasler, Nov 22 2025

Crossrefs

Essentially the same as A000918 shifted, and A095121.

Cf. A307549.

Sequence in context: A390083 A394990 A395004 * A192672 A396751 A175656

Adjacent sequences: A307454 A307455 A307456 * A307458 A307459 A307460

Keyword

nonn,easy

Author

Eric W. Weisstein, Apr 08 2019

Extensions

a(5) onwards from Andrew Howroyd, Jun 09 2025

Status

approved