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A180576 Wiener index of the n-web graph. 1

%I #21 Mar 29 2020 13:10:17

%S 4,27,69,148,255,417,616,888,1206,1615,2079,2652,3289,4053,4890,5872,

%T 6936,8163,9481,10980,12579,14377,16284,18408,20650,23127,25731,28588,

%U 31581,34845,38254,41952,45804,49963,54285,58932,63751,68913,74256,79960

%N Wiener index of the n-web graph.

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

%C 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.

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

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

%C Sequence extended to a(1)-a(2) using the formula/recurrence. - _Eric W. Weisstein_, Sep 08 2017

%H B. E. Sagan, Y-N. Yeh and P. Zhang, <a href="http://users.math.msu.edu/users/sagan/Papers/Old/wpg-pub.pdf">The Wiener Polynomial of a Graph</a>, Internat. J. of Quantum Chem., 60, 1996, 959-969.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WebGraph.html">Web Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WienerIndex.html">Wiener Index</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-4,1,2,-1).

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

%F 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

%F a(n) = n*(2*n*(9*n+40)+9*(-1)^n-25)/16. - _Bruno Berselli_, Oct 31 2012

%p 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);

%t Table[n (-25 + 9 (-1)^n + 2 n (40 + 9 n))/16, {n, 20}] (* _Eric W. Weisstein_, Sep 08 2017 *)

%t LinearRecurrence[{2, 1, -4, 1, 2, -1}, {4, 27, 69, 148, 255, 4178}, 20] (* _Eric W. Weisstein_, Sep 08 2017 *)

%t 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 *)

%Y Cf. A180575.

%K nonn,easy

%O 1,1

%A _Emeric Deutsch_, Sep 19 2010

%E a(1)-a(2) from _Eric W. Weisstein_, Sep 08 2017

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Last modified March 29 03:51 EDT 2024. Contains 371264 sequences. (Running on oeis4.)