A349416 a(n) is the Wiener index of a broom on 2n vertices of which n+2 are pendant.

25, 54, 100, 167, 259, 380, 534, 725, 957, 1234, 1560, 1939, 2375, 2872, 3434, 4065, 4769, 5550, 6412, 7359, 8395, 9524, 10750, 12077, 13509, 15050, 16704, 18475, 20367, 22384, 24530, 26809, 29225, 31782, 34484, 37335, 40339, 43500, 46822, 50309, 53965, 57794, 61800, 65987

Offset

3,1

Comments

A broom on 2n vertices is a caterpillar that is obtained by adding n pendant vertices to the first (or last) internal vertex of a path on n >= 3 vertices.

Links

Table of n, a(n) for n=3..46.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

a(n) = 2*n^3/3 + n^2/2 + 5*n/6.

From Elmo R. Oliveira, May 30 2026: (Start)

G.f.: x^3*(25 - 46*x + 34*x^2 - 9*x^3)/(1 - x)^4.

E.g.f.: x*(exp(x)*(12 + 15*x + 4*x^2) - (12 + 27*x))/6.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 6. (End)

Example

For n=3 the value a(3)=25 gives the Wiener index of a star graph on 6 vertices. For n=4, a(4)=54 gives the Wiener index of a broom graph on 8 vertices (6 of which are leaves).
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Mathematica

nterms=50; Table[2n^3/3+n^2/2+5n/6, {n, 3, nterms+2}] (* Paolo Xausa, Nov 22 2021 *)

Crossrefs

Cf. A349417 (sling), A349418 (tridon).

Sequence in context: A117472 A146633 A123825 * A157269 A371129 A186892

Adjacent sequences: A349413 A349414 A349415 * A349417 A349418 A349419

Keyword

nonn,easy

Author

Julian Allagan, Nov 16 2021

Status

approved