A126184 Number of hex trees with n edges and having no nonroot nodes of outdegree 2.

1, 3, 10, 33, 108, 351, 1134, 3645, 11664, 37179, 118098, 373977, 1180980, 3720087, 11691702, 36669429, 114791256, 358722675, 1119214746, 3486784401, 10847773692, 33705582543, 104603532030, 324270949293, 1004193907488, 3106724901291, 9602604240354, 29655101330505

Offset

0,2

Comments

A hex tree is a rooted tree where each vertex has 0, 1, or 2 children and, when only one child is present, it is either a left child, or a middle child, or a right child (name due to an obvious bijection with certain tree-like polyhexes; see the Harary-Read reference).

Links

Table of n, a(n) for n=0..27.

Frank Harary and Ronald C. Read, The enumeration of tree-like polyhexes, Proc. Edinburgh Math. Soc. (2) 17 (1970), 1-13.

Index entries for linear recurrences with constant coefficients, signature (6,-9).

Formula

a(n) = A126183(n,0).

a(n) = (n+8)*3^(n-2) for n >= 1; a(0)=1.

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

From Paul Curtz, Mar 27 2022: (Start)

a(n+1) = 3*a(n) + A140429(n), for n >= 0; a(0)=1.

Binomial transform of A172481(n) for n >= 0.

Also, with a different offset, the binomial transform of A045891(n+2) for n >= 0. (End)

From Amiram Eldar, Dec 09 2025: (Start)

Sum_{n>=0} 1/a(n) = 59049*log(3/2) - 6703433/280.

Sum_{n>=0} (-1)^n/a(n) = 4756663/280 - 59049*log(4/3). (End)

Maple

1, seq(3^(n-2)*(n+8), n=1..28);

Mathematica

a[n_] := (n+8) * 3^(n-2); a[0] = 1; Array[a, 50, 0] (* Amiram Eldar, Dec 09 2025 *)

Crossrefs

Cf. A126183.

Cf. A045891, A140429, A172481.

Sequence in context: A120897 A077825 A049219 * A292397 A060557 A018920

Adjacent sequences: A126181 A126182 A126183 * A126185 A126186 A126187

Keyword

nonn,easy

Author

Emeric Deutsch, Dec 19 2006

Status

approved