A143941 The Wiener index of a chain of n triangles (i.e., joined like VVV..VV; here V is a triangle!).

3, 14, 37, 76, 135, 218, 329, 472, 651, 870, 1133, 1444, 1807, 2226, 2705, 3248, 3859, 4542, 5301, 6140, 7063, 8074, 9177, 10376, 11675, 13078, 14589, 16212, 17951, 19810, 21793, 23904, 26147, 28526, 31045, 33708, 36519, 39482, 42601, 45880, 49323, 52934

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.

Row 2 of the convolution array A213752. - Clark Kimberling, Jun 20 2012

Also the circuit rank of the (n+2) X (n+2) bishop graph. - Eric W. Weisstein, May 10 2019

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Eric Weisstein's World of Mathematics, Bishop Graph

Eric Weisstein's World of Mathematics, Circuit Rank

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

Formula

a(n) = n*(1 + 6*n + 2*n^2)/3.

G.f.: z*(3 + 2*z - z^2)/(1-z)^4.

a(n) = Sum_{k=1..n} k*A143940(n,k).

a(n) = Sum_{k=1..n} A142463(k). - Richard R. Forberg, Jan 09 2015

a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Wesley Ivan Hurt, Apr 08 2015

E.g.f.: exp(x)*x*(9 + 12*x + 2*x^2)/3. - Stefano Spezia, Jan 03 2022

Example

a(2)=14 because in the graph VV (V is a triangle!) we have 6 distances equal to 1 and 4 distances equal to 2.

Maple

seq((1/3)*n*(1+6*n+2*n^2), n=1..43);

Mathematica

CoefficientList[Series[(3+2*x-x^2)/(1-x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jul 03 2012 *)
LinearRecurrence[{4, -6, 4, -1}, {3, 14, 37, 76}, 50] (* Harvey P. Dale, Sep 06 2023 *)

Prog

(Magma) [n*(1+6*n+2*n^2)/3 : n in [1..40]]; // Wesley Ivan Hurt, Apr 08 2015

Crossrefs

Cf. A142463, A143940, A213752.

Sequence in context: A179132 A068044 A141129 * A162147 A319791 A027444

Adjacent sequences: A143938 A143939 A143940 * A143942 A143943 A143944

Keyword

nonn,easy

Author

Emeric Deutsch, Sep 06 2008

Status

approved