A138179 Wiener index of the prism graph Y_n on 2n nodes.

1, 8, 21, 48, 85, 144, 217, 320, 441, 600, 781, 1008, 1261, 1568, 1905, 2304, 2737, 3240, 3781, 4400, 5061, 5808, 6601, 7488, 8425, 9464, 10557, 11760, 13021, 14400, 15841, 17408, 19041, 20808, 22645, 24624, 26677, 28880, 31161, 33600, 36121, 38808

Offset

1,2

Comments

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

Apparently a(n) = n * A074148(n), so a(n)= +2*a(n-1) +a(n-2) -4*a(n-3) +a(n-4) +2*a(n-5) -a(n-6). - R. J. Mathar, May 31 2010

From Emeric Deutsch, Sep 16 2010: (Start)

The Wiener index of a connected graph is the sum of all distances in the graph.

Y_n is also called circular ladder (= P_2 X C_n, where P_2 is the path graph on 2 nodes and C_n is the cycle graph on n nodes).

a(n) = Sum(k*A180572(n,k), k>=1).

a(n) is the derivative of the Wiener polynomial of Y_n (given in A180572) evaluated at t=1. (see the Sagan et al. reference).

(End)

References

J. Gross and J. Yellen, Graph Theory and its Applications, CRC, Boca Raton, 1999 (p. 14). - Emeric Deutsch, Sep 16 2010

Links

Colin Barker, Table of n, a(n) for n = 1..1000 (corrected by Michel Marcus, Jan 19 2019)

B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969. - Emeric Deutsch, Sep 16 2010

Y.-N. Yeh and I. Gutman, On the sum of all distances in composite graphs, Discrete Math., 135 (1994), 359-365 (set m=2 in the formula for W(Cyl_{m,n}) on p. 363). - Emeric Deutsch, Sep 16 2010

Eric Weisstein's World of Mathematics, Prism 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

From Emeric Deutsch, Sep 16 2010: (Start)

a(2n+1) = (2n+1)(2n^2+4*n+1); a(2n)=4n^2*(n+1).

G.f.: (z (1 + 6 z + 4 z^2 + 2 z^3 - z^4))/((-1 + z)^4 (1 + z)^2).

(End)

a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6).

Example

a(3) = 21 because the triangular prism has 9 distances equal to 1 (the edges) and 6 distances equal to 2 (from the vertices of the lower base to the "opposite" vertices of the upper base). - Emeric Deutsch, Sep 16 2010

Mathematica

LinearRecurrence[{2, 1, -4, 1, 2, -1}, {1, 8, 21, 48, 85, 144}, 40] (* Harvey P. Dale, Jul 29 2013 *)
Table[1/4 n (-1 + (-1)^n + 2 n (2 + n)), {n, 20}] (* Eric W. Weisstein, May 11 2017 *)
CoefficientList[Series[(1 + 6 x + 4 x^2 + 2 x^3 - x^4)/((-1 + x)^4 (1 + x)^2), {x, 0, 20}], x] (* Eric W. Weisstein, Sep 08 2017 *)

Prog

(PARI) Vec((x*(1+ 6*x+4*x^2+2*x^3-x^4))/((-1+x)^4*(1+x)^2) + O(x^50)) \\ Colin Barker, Jun 23 2015; Michel Marcus, Jan 19 2019

Crossrefs

Cf. A180572

Sequence in context: A273602 A258448 A344599 * A067334 A066859 A227653

Adjacent sequences: A138176 A138177 A138178 * A138180 A138181 A138182

Keyword

nonn,easy

Author

Eric W. Weisstein, Mar 04 2008

Extensions

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

Status

approved