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A180579 The Wiener index of the Dutch windmill graph D(5,n) (n>=1).

%I A180579 #24 Apr 03 2025 16:02:57
%S A180579 15,78,189,348,555,810,1113,1464,1863,2310,2805,3348,3939,4578,5265,
%T A180579 6000,6783,7614,8493,9420,10395,11418,12489,13608,14775,15990,17253,
%U A180579 18564,19923,21330,22785,24288,25839,27438,29085,30780,32523,34314,36153,38040,39975,41958,43989
%N A180579 The Wiener index of the Dutch windmill graph D(5,n) (n>=1).
%C A180579 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). The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph.
%H A180579 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 A180579 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DutchWindmillGraph.html">Dutch Windmill Graph</a>.
%H A180579 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A180579 a(n) = 3*n*(8*n-3).
%F A180579 a(n) = A180867(4,n).
%F A180579 The Wiener polynomial of the graph D(5,n) is nt(t+1)[2(n-1)t^2+2(n-1)t+5].
%F A180579 G.f.: -3*x*(11*x+5)/(x-1)^3. - _Colin Barker_, Oct 31 2012
%F A180579 From _Elmo R. Oliveira_, Apr 03 2025: (Start)
%F A180579 E.g.f.: 3*exp(x)*x*(5 + 8*x).
%F A180579 a(n) = 3*A139273(n).
%F A180579 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)
%e A180579 a(1)=15 because in D(5,1)=C_5 we have 5 distances equal to 1 and 5 distances equal to 2.
%p A180579 seq(3*n*(8*n-3), n = 1 .. 40);
%t A180579 Table[3n(8n-3),{n,40}] (* or *) LinearRecurrence[{3,-3,1},{15,78,189},40] (* _Harvey P. Dale_, May 01 2023 *)
%o A180579 (PARI) a(n)=3*n*(8*n-3) \\ _Charles R Greathouse IV_, Jun 17 2017
%Y A180579 Cf. A014642, A033991, A139273, A180578, A180867.
%K A180579 nonn,easy
%O A180579 1,1
%A A180579 _Emeric Deutsch_, Sep 30 2010
%E A180579 More terms from _Elmo R. Oliveira_, Apr 03 2025