A143943 The Wiener index of a chain of n squares joined at vertices (i.e., joined like <><><>...<>; here <> is a square!).

8, 40, 114, 248, 460, 768, 1190, 1744, 2448, 3320, 4378, 5640, 7124, 8848, 10830, 13088, 15640, 18504, 21698, 25240, 29148, 33440, 38134, 43248, 48800, 54808, 61290, 68264, 75748, 83760, 92318, 101440, 111144, 121448, 132370, 143928, 156140, 169024, 182598, 196880

Offset

1,1

Comments

The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph.

Links

Paolo Xausa, Table of n, a(n) for n = 1..10000

T. Mansour and M. Schork, Wiener, hyper-Wiener, detour and hyper-detour indices of bridge and chain graphs, J. Math. Chemistry, 47, 2010, 72-98 (see Example 5.6).

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

Formula

a(n) = n*(2 + 3*n + 3*n^2).

G.f.: 2*z*(2 + z)^2/(1 - z)^4.

a(n) = Sum_{k=1..2*n} k*A143942(n,k).

From Enrique Navarrete, Feb 25 2026: (Start)

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

E.g.f.: exp(x)*(3*x^3 + 12*x^2 + 8*x). (End)

Example

a(1)=8 because in the graph <> with vertices a,b,c,d we have 4 distances equal to 1 (the edges) and 2 distances equal to 2 (ac and bd); 4*1 + 2*2 = 8.

Maple

seq(n*(2+3*n+3*n^2), n=1..40);

Mathematica

A143943[n_] := n*(3*n*(n+1) + 2); Array[A143943, 50] (* or *)
LinearRecurrence[{4, -6, 4, -1}, {8, 40, 114, 248}, 50] (* Paolo Xausa, Mar 30 2026 *)

Crossrefs

Cf. A143942.

Sequence in context: A383465 A211631 A279273 * A135796 A105374 A162668

Adjacent sequences: A143940 A143941 A143942 * A143944 A143945 A143946

Keyword

nonn,easy

Author

Emeric Deutsch, Sep 06 2008

Status

approved