A180578 -id:A180578 - cp's OEIS frontend

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

15, 78, 189, 348, 555, 810, 1113, 1464, 1863, 2310, 2805, 3348, 3939, 4578, 5265, 6000, 6783, 7614, 8493, 9420, 10395, 11418, 12489, 13608, 14775, 15990, 17253, 18564, 19923, 21330, 22785, 24288, 25839, 27438, 29085, 30780, 32523, 34314, 36153, 38040, 39975, 41958, 43989

Offset: 1

Emeric Deutsch, Sep 30 2010

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.

			a(1)=15 because in D(5,1)=C_5 we have 5 distances equal to 1 and 5 distances equal to 2.

B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., Vol. 60, 1996, pp. 959-969.
Eric Weisstein's World of Mathematics, Dutch Windmill Graph.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A014642, A033991, A139273, A180578, A180867.

Maple
```
seq(3*n*(8*n-3), n = 1 .. 40);
```

Table[3n(8n-3),{n,40}] (* or *) LinearRecurrence[{3,-3,1},{15,78,189},40] (* Harvey P. Dale, May 01 2023 *)

a(n)=3*n*(8*n-3) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 3*n*(8*n-3).
a(n) = A180867(4,n).
The Wiener polynomial of the graph D(5,n) is nt(t+1)[2(n-1)t^2+2(n-1)t+5].
G.f.: -3*x*(11*x+5)/(x-1)^3. - Colin Barker, Oct 31 2012
From Elmo R. Oliveira, Apr 03 2025: (Start)
E.g.f.: 3*exp(x)*x*(5 + 8*x).
a(n) = 3*A139273(n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

More terms from Elmo R. Oliveira, Apr 03 2025

A180867 Square array read by antidiagonals: T(m,n) is the Wiener index of the Dutch windmill graph D(m,n) (m>=3, n>=1).

Original entry on oeis.org

3, 8, 14, 15, 40, 33, 27, 78, 96, 60, 42, 144, 189, 176, 95, 64, 228, 351, 348, 280, 138, 90, 352, 558, 648, 555, 408, 189, 125, 500, 864, 1032, 1035, 810, 560, 248, 165, 700, 1230, 1600, 1650, 1512, 1113, 736, 315, 216, 930, 1725, 2280, 2560, 2412, 2079, 1464

Offset: 3

Emeric Deutsch, Sep 30 2010

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.

			Square array starts:
   3,  14,  33,  60,   95, ...
   8,  40,  96, 176,  280, ...
  15,  78, 189, 348,  555, ...
  27, 144, 351, 648, 1035, ...

B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.
Eric Weisstein's World of Mathematics, Dutch Windmill Graph.

Cf. A014642, A033991, A180578, A180579.

T := proc (m, n) if `mod`(m, 2) = 1 then (1/8)*n*(m^2-1)*(2*m*n-m-2*n+2) else (1/8)*n*m^2*(2*m*n-m-2*n+2) end if end proc: for n from 3 to 12 do seq(T(n+1-j, j), j = 1 .. n-2) end do; # yields sequence in triangular form

T(3,n) = A033991(n).
T(4,n) = A014642(n).
T(5,n) = A180579(n).
T(6,n) = A180578(n).
T(m,n) = (1/8)n(m^2-1)(2mn-m-2n+2) if m is odd.
T(m,n) = (1/8)n*m^2*(2mn-m-2n+2) if m is even.
The Wiener polynomial of D(m,n) is n*w(m)+(1/2)n(n-1)r(m)^2, where, denoting S(t,p) = Sum_{j=1..p-1}t^j, we have w(m) = mS(t,m/2) + (1/2)mt^(m/2), r(m) = 2S(t,m/2) + t^(m/2) if m is even and w(m) = mS(t,(m+1)/2), r(m) = 2S(t,(m+1)/2) if m is odd.

Typos in Wiener polynomial information corrected by Emeric Deutsch, Sep 30 2010

cp's OEIS Frontend

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

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