A180576 Wiener index of the n-web graph.

4, 27, 69, 148, 255, 417, 616, 888, 1206, 1615, 2079, 2652, 3289, 4053, 4890, 5872, 6936, 8163, 9481, 10980, 12579, 14377, 16284, 18408, 20650, 23127, 25731, 28588, 31581, 34845, 38254, 41952, 45804, 49963, 54285, 58932, 63751, 68913, 74256, 79960

Offset: 1

Author

Emeric Deutsch, Sep 19 2010

Keywords

• nonn
• easy

Comments

The n-web graph is the stacked prism graph C_n X P_3 with the edges of the outer cycle removed.

Equivalently, the n-web graph is obtained by attaching a pendant edge to each node of the outer cycle of the circular ladder (prism) C_n X P_2.

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

a(n) = sum(A180575(n,k), k>=1).

Sequence extended to a(1)-a(2) using the formula/recurrence. - Eric W. Weisstein, Sep 08 2017

Links

• 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, Web Graph
• Eric Weisstein's World of Mathematics, Wiener Index
• Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Crossrefs

Cf. A180575.

Programs

Maple

a := proc (n) if `mod`(n, 2) = 1 then (1/8)*n*(9*n^2+40*n-17) else (1/8)*n*(9*n^2+40*n-8) end if end proc: seq(a(n), n = 3 .. 45);

Mathematica

Table[n (-25 + 9 (-1)^n + 2 n (40 + 9 n))/16, {n, 20}] (* Eric W. Weisstein, Sep 08 2017 *)
LinearRecurrence[{2, 1, -4, 1, 2, -1}, {4, 27, 69, 148, 255, 4178}, 20] (* Eric W. Weisstein, Sep 08 2017 *)
CoefficientList[Series[(4 + 19 x + 11 x^2 - x^3 - 6 x^4 + 3761 x^5)/((-1 + x)^4 (1 + x)^2), {x, 0, 20}], x] (* Eric W. Weisstein, Sep 08 2017 *)

Formula

a(2n) = n*(9*n^2+20*n-2); a(2n+1) = (2*n+1)*(9*n^2+29*n+8)/2.

G.f.: -x^3*(27*x^5-50*x^4-35*x^3+110*x^2-10*x-69)/((x-1)^4*(x+1)^2). - Colin Barker, Oct 31 2012

a(n) = n*(2*n*(9*n+40)+9*(-1)^n-25)/16. - Bruno Berselli, Oct 31 2012

Extensions

a(1)-a(2) from Eric W. Weisstein, Sep 08 2017