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

0, 0, 3, 21, 75, 195, 420, 798, 1386, 2250, 3465, 5115, 7293, 10101, 13650, 18060, 23460, 29988, 37791, 47025, 57855, 70455, 85008, 101706, 120750, 142350, 166725, 194103, 224721, 258825, 296670, 338520, 384648, 435336, 490875, 551565, 617715, 689643

Offset: 1

Author

Emeric Deutsch, Aug 26 2013

Keywords

• nonn
• easy

Comments

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.

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).

References

• R. A. Brualdi and H. J. Ryser, Combinatorial Matrix Theory, Cambridge Univ. Press, 1992.

Links

• G. G. Cash, Relationship between the Hosoya polynomial and the hyper-Wiener index, Applied Mathematics Letters, 15(7) (2002), 893-895.
• Eric Weisstein's World of Mathematics, Triangular Graph.
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A001296, A006011.

Programs

Maple

a := proc (n) options operator, arrow: (1/8)*n*(n-1)*(n-2)*(3*n-5) end proc: seq(a(n), n = 1 .. 38);

Mathematica

LinearRecurrence[{5,-10,10,-5,1},{0,0,3,21,75},40] (* Harvey P. Dale, Feb 23 2023 *)

Formula

a(n) = n*(n - 1)*(n - 2)*(3*n - 5)/8.

G.f.: 3*x^3*(1 + 2*x)/(1 - x)^5.

The Hosoya-Wiener polynomial of T(n) is (1/8)*n*(n - 1)*(4 + 4*(n-2)*t + (n - 2)*(n - 3)*t^2).

a(n) = 3*A001296(n-2) for n >= 2. - R. J. Mathar, Mar 05 2017