OFFSET
1,3
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).
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
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 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, Aug 26 2013
STATUS
approved