OFFSET
5,1
COMMENTS
In other words, the number of unordered pairs of triangles that share at most 1 vertex in the complete graph K_n.
LINKS
Julian D. Allagan, Enumeration and asymptotic analysis of edge-disjoint triangle packings in complete graphs, Journal of Combinatorial Mathematics and Combinatorial Computing, Volume 128, 3-30, 2025. See p. 17.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).
FORMULA
a(n) = 10*binomial(n,6) + 3*n*binomial(n-1,4).
a(n) = n*(n-1)*(n-2)*(n-3)*(n-4)*(n+4)/72.
G.f.: 5*x^5*(3 - x)/(1 - x)^7. - Stefano Spezia, Mar 09 2025
From Amiram Eldar, Oct 25 2025: (Start)
Sum_{n>=5} 1/a(n) = 6343/78400.
Sum_{n>=5} (-1)^(n+1)/a(n) = 288*log(2)/35 - 442567/78400. (End)
EXAMPLE
a(5) = 15 because there are 15 unordered pairs of triangles that share 1 vertex.
a(6) = 100 = 90 + 10 because there are 90 = 15*binomial(6,5) unordered pairs of triangles that share 1 vertex and 10 = 10*binomial(6,6) unordered pairs of triangles that do not share a vertex.
MATHEMATICA
a[n_]:=n*(n-1)*(n-2)*(n-3)*(n-4)*(n+4)/72; Array[a, 33, 5] (* Stefano Spezia, Mar 09 2025 *)
PROG
(Python)
def A381862(n): return n*(n*(n*(n*(n*(n-6)-5)+90)-176)+96)//72 # Chai Wah Wu, Mar 18 2025
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Julian Allagan, Mar 08 2025
STATUS
approved