cp's OEIS Frontend

A180578 The Wiener index of the Dutch windmill graph D(6,n) (n>=1).

%I A180578 #24 Apr 14 2025 17:31:33
%S A180578 27,144,351,648,1035,1512,2079,2736,3483,4320,5247,6264,7371,8568,
%T A180578 9855,11232,12699,14256,15903,17640,19467,21384,23391,25488,27675,
%U A180578 29952,32319,34776,37323,39960,42687,45504,48411,51408,54495,57672,60939,64296,67743,71280,74907
%N A180578 The Wiener index of the Dutch windmill graph D(6,n) (n>=1).
%C A180578 The Dutch windmill graph D(m,n) (also called friendship graph) is the graph obtained by taking n copies of the cycle graph C_m with a vertex in common (i.e., a bouquet of n C_m graphs).
%C A180578 The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph.
%H A180578 B. E. Sagan, Y-N. Yeh and P. Zhang, <a href="http://users.math.msu.edu/users/sagan/Papers/Old/wpg-pub.pdf">The Wiener Polynomial of a Graph</a>, Internat. J. of Quantum Chem., Vol. 60, 1996, pp. 959-969.
%H A180578 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DutchWindmillGraph.html">Dutch Windmill Graph</a>.
%H A180578 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A180578 a(n) = A180867(6,n).
%F A180578 a(n) = 9*n*(5*n-2).
%F A180578 The Wiener polynomial of the graph D(6,n) is (1/2)nt(t^2+2t+2)((n-1)t^3+2(n-1)t^2+2(n-1)t+6).
%F A180578 G.f.: -9*x*(7*x+3)/(x-1)^3. - _Colin Barker_, Oct 31 2012
%F A180578 From _Elmo R. Oliveira_, Apr 03 2025: (Start)
%F A180578 E.g.f.: 9*exp(x)*x*(3 + 5*x).
%F A180578 a(n) = 9*A147874(n+1).
%F A180578 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)
%e A180578 a(1)=27 because in D(6,1)=C_6 we have 6 distances equal to 1, 6 distances equal to 2, and 3 di stances equal to 3.
%p A180578 seq(9*n*(5*n-2), n = 1 .. 40);
%o A180578 (PARI) a(n)=9*n*(5*n-2) \\ _Charles R Greathouse IV_, Jun 17 2017
%Y A180578 Cf. A014642, A033991, A147874, A180579, A180867.
%K A180578 nonn,easy
%O A180578 1,1
%A A180578 _Emeric Deutsch_, Sep 30 2010
%E A180578 More terms from _Elmo R. Oliveira_, Apr 03 2025
%E A180578 Duplicated a(38) removed by _Sean A. Irvine_, Apr 14 2025