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
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, Sep 06 2008
STATUS
approved