cp's OEIS Frontend

A228317 The hyper-Wiener index of the triangular graph T(n) (n >= 1).

%I A228317 #20 Feb 16 2025 08:33:20
%S A228317 0,0,3,21,75,195,420,798,1386,2250,3465,5115,7293,10101,13650,18060,
%T A228317 23460,29988,37791,47025,57855,70455,85008,101706,120750,142350,
%U A228317 166725,194103,224721,258825,296670,338520,384648,435336,490875,551565,617715,689643
%N A228317 The hyper-Wiener index of the triangular graph T(n) (n >= 1).
%C A228317 The triangular graph T(n) is the graph whose vertices represent the 2-subsets of {1,2,...,n} and two vertices are adjacent provided the corresponding 2-subsets have a nonempty intersection.
%C A228317 The triangular graph T(n) is a strongly regular graph with parameters n*(n-1)/2, 2*(n-2), n-2, and 4 (see the Brualdi and Ryser reference, Theorem 5.2.4).
%D A228317 R. A. Brualdi and H. J. Ryser, Combinatorial Matrix Theory, Cambridge Univ. Press, 1992.
%H A228317 G. G. Cash, <a href="https://doi.org/10.1016/S0893-9659(02)00059-9">Relationship between the Hosoya polynomial and the hyper-Wiener index</a>, Applied Mathematics Letters, 15(7) (2002), 893-895.
%H A228317 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TriangularGraph.html">Triangular Graph</a>.
%H A228317 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A228317 a(n) = n*(n - 1)*(n - 2)*(3*n - 5)/8.
%F A228317 G.f.: 3*x^3*(1 + 2*x)/(1 - x)^5.
%F A228317 The Hosoya-Wiener polynomial of T(n) is (1/8)*n*(n - 1)*(4 + 4*(n-2)*t + (n - 2)*(n - 3)*t^2).
%F A228317 a(n) = 3*A001296(n-2) for n >= 2. - _R. J. Mathar_, Mar 05 2017
%p A228317 a := proc (n) options operator, arrow: (1/8)*n*(n-1)*(n-2)*(3*n-5) end proc: seq(a(n), n = 1 .. 38);
%t A228317 LinearRecurrence[{5,-10,10,-5,1},{0,0,3,21,75},40] (* _Harvey P. Dale_, Feb 23 2023 *)
%Y A228317 Cf. A001296, A006011.
%K A228317 nonn,easy
%O A228317 1,3
%A A228317 _Emeric Deutsch_, Aug 26 2013