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

1,1

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

Table of n, a(n) for n=1..40.

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).

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

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 *)

Crossrefs

Cf. A180575.

Sequence in context: A175701 A227866 A356814 * A158186 A272858 A272818

Adjacent sequences: A180573 A180574 A180575 * A180577 A180578 A180579

Keyword

nonn,easy

Author

Emeric Deutsch, Sep 19 2010

Extensions

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

Status

approved