This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A138179 #34 Feb 16 2025 08:33:07
%S A138179 1,8,21,48,85,144,217,320,441,600,781,1008,1261,1568,1905,2304,2737,
%T A138179 3240,3781,4400,5061,5808,6601,7488,8425,9464,10557,11760,13021,14400,
%U A138179 15841,17408,19041,20808,22645,24624,26677,28880,31161,33600,36121,38808
%N A138179 Wiener index of the prism graph Y_n on 2n nodes.
%C A138179 Sequence expended to a(1)-a(2) using the formula/recurrence. - _Eric W. Weisstein_, Sep 08 2017
%C A138179 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
%C A138179 From _Emeric Deutsch_, Sep 16 2010: (Start)
%C A138179 The Wiener index of a connected graph is the sum of all distances in the graph.
%C A138179 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).
%C A138179 a(n) = Sum(k*A180572(n,k), k>=1).
%C A138179 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).
%C A138179 (End)
%D A138179 J. Gross and J. Yellen, Graph Theory and its Applications, CRC, Boca Raton, 1999 (p. 14). - _Emeric Deutsch_, Sep 16 2010
%H A138179 Colin Barker, <a href="/A138179/b138179.txt">Table of n, a(n) for n = 1..1000</a> (corrected by Michel Marcus, Jan 19 2019)
%H A138179 B. E. Sagan, Y-N. Yeh and P. Zhang, <a href="http://dx.doi.org/10.1002/(SICI)1097-461X(1996)60:5<959::AID-QUA2>3.0.CO;2-W">The Wiener Polynomial of a Graph</a>, Internat. J. of Quantum Chem., 60, 1996, 959-969. - _Emeric Deutsch_, Sep 16 2010
%H A138179 Y.-N. Yeh and I. Gutman, <a href="http://dx.doi.org/10.1016/0012-365X(93)E0092-I">On the sum of all distances in composite graphs</a>, 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
%H A138179 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrismGraph.html">Prism Graph</a>
%H A138179 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WienerIndex.html">Wiener Index</a>
%H A138179 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-4,1,2,-1).
%F A138179 From _Emeric Deutsch_, Sep 16 2010: (Start)
%F A138179 a(2n+1) = (2n+1)(2n^2+4*n+1); a(2n)=4n^2*(n+1).
%F A138179 G.f.: (z (1 + 6 z + 4 z^2 + 2 z^3 - z^4))/((-1 + z)^4 (1 + z)^2).
%F A138179 (End)
%F A138179 a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6).
%e A138179 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
%t A138179 LinearRecurrence[{2, 1, -4, 1, 2, -1}, {1, 8, 21, 48, 85, 144}, 40] (* _Harvey P. Dale_, Jul 29 2013 *)
%t A138179 Table[1/4 n (-1 + (-1)^n + 2 n (2 + n)), {n, 20}] (* _Eric W. Weisstein_, May 11 2017 *)
%t A138179 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 *)
%o A138179 (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
%Y A138179 Cf. A180572
%K A138179 nonn,easy
%O A138179 1,2
%A A138179 _Eric W. Weisstein_, Mar 04 2008
%E A138179 a(1)-a(2) from _Eric W. Weisstein_, Sep 08 2017