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URL: https://oeis.org/A138179

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A138179
Wiener index of the prism graph Y_n on 2n nodes.
5
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
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
Sequence in context: A273602 A258448 A344599 * A067334 A066859 A227653
KEYWORD
nonn,easy
AUTHOR
Eric W. Weisstein, Mar 04 2008
EXTENSIONS
a(1)-a(2) from Eric W. Weisstein, Sep 08 2017
STATUS
approved