cp's OEIS Frontend

A192025 The Wiener index of the double-comb graph \/\/\/...\/_\/ with 3n (n>=1) nodes. The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices in the graph.

%I A192025 #20 Aug 07 2025 13:18:15
%S A192025 4,29,84,178,320,519,784,1124,1548,2065,2684,3414,4264,5243,6360,7624,
%T A192025 9044,10629,12388,14330,16464,18799,21344,24108,27100,30329,33804,
%U A192025 37534,41528,45795,50344,55184,60324,65773,71540,77634,84064,90839,97968,105460
%N A192025 The Wiener index of the double-comb graph \/_\/_\/...\/_\/ with 3n (n>=1) nodes. The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices in the graph.
%C A192025 a(n) = Sum(k*A192024(n,k),k>=1).
%H A192025 T. Mansour, M. Schork, <a href="http://dx.doi.org/10.1016/j.dam.2008.09.008">The vertex PI index and Szeged index of bridge graphs</a>, Discrete Appl. Math., 157, 2009, 1600-1606 (see last page).
%H A192025 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A192025 a(n) = n*(3*n^2+12*n-7)/2.
%F A192025 G.f.: x*(4+13*x-8*x^2)/(1-x)^4.
%e A192025 a(2)=29 because in the graph \/_\/ there are 5 pairs of nodes at distance 1, 6 pairs at distance 2, and 4 pairs at distance 3 (5*1 + 6*2 + 4*3 = 29).
%p A192025 a := n -> (1/2)*n*(3*n^2+12*n-7): seq(a(n), n = 1 .. 40);
%t A192025 LinearRecurrence[{4,-6,4,-1},{4,29,84,178},40] (* _Harvey P. Dale_, Aug 07 2025 *)
%Y A192025 Cf. A192024
%K A192025 nonn,easy
%O A192025 1,1
%A A192025 _Emeric Deutsch_, Jun 25 2011