A143366 Wiener index of the hexagon crown (beehive model) with n hexagons on each side of the outside ring.

27, 1002, 7809, 33204, 101751, 253758, 549213, 1071720, 1932435, 3274002, 5274489, 8151324, 12165231, 17624166, 24887253, 34368720, 46541835, 61942842, 81174897, 104912004, 133902951, 168975246, 211039053, 261091128, 320218755

Offset: 1

Author

Emeric Deutsch, Sep 04 2008

Keywords

• nonn

Comments

The Wiener index of a connected graph G is the sum of distances between all pairs of vertices in G.

The second Maple program finds and sums all the binomial(24,2) distances.

Examples

			a(1)=27 because in the hexagon ABCDEF the binomial(6,2)=15 distances are AB=BC=CD=DE=EF=FA=1, AC=BD=CE=DF=EA=FB=2, AD=BE=CF=3 and their sum is 27.
		

Links

• Bo-Yin Yang and Yeong-Nan Yeh, A Crowning Moment for Wiener Indices, Studies in Appl. Math., 112 (2004), 333-340.
• Index entries for linear recurrences with constant coefficients, signature (6, -15, 20, -15, 6, -1).

Programs

Maple

W:=proc(n) options operator, arrow: (164/5)*n^5-6*n^3+(1/5)*n end proc; seq(W(n),n=1..25);
with(GraphTheory); G := Graph([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], {{1, 2}, {2, 3}, {3, 4}, {4, 5}, {5, 6}, {6, 7}, {7, 8}, {8, 9}, {9, 10}, {10, 11}, {11, 12}, {12, 13}, {13, 14}, {14, 15}, {15, 16}, {16, 17}, {17, 18}, {1, 18}, {2, 19}, {5, 20}, {8, 21}, {11, 22}, {14, 23}, {17, 24}, {19, 20}, {20, 21}, {21, 22}, {22, 23}, {23, 24}, {19, 24}}); d := AllPairsDistance(G); with(LinearAlgebra); n := 24; add(add(d[i, j], j = i .. n), i = 1 .. n);

Mathematica

Table[n (164n^4-30n^2+1)/5,{n,30}] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{27,1002,7809,33204,101751,253758},30] (* Harvey P. Dale, Jun 09 2024 *)

PARI

a(n) = n*(164*n^4-30*n^2+1)/5; \\ Michel Marcus, Jan 17 2019

Formula

a(n) = n(164n^4-30n^2+1)/5.

G.f.: 3x(9+280x+734x^2+280x^3+9x^4)/(1-x)^6. [R. J. Mathar, Sep 05 2008]