cp's OEIS Frontend

A381862 Number of pairs of triangles that are pairwise edge-disjoint in the complete graph K_n.

%I A381862 #45 Mar 28 2025 08:35:45
%S A381862 15,100,385,1120,2730,5880,11550,21120,36465,60060,95095,145600,
%T A381862 216580,314160,445740,620160,847875,1141140,1514205,1983520,2567950,
%U A381862 3289000,4171050,5241600,6531525,8075340,9911475,12082560,14635720,17622880,21101080,25132800,29786295
%N A381862 Number of pairs of triangles that are pairwise edge-disjoint in the complete graph K_n.
%C A381862 In other words, the number of unordered pairs of triangles that share at most 1 vertex in the complete graph K_n.
%D A381862 Julian Allagan, Edge-Disjoint Triangle Packings in Complete Graphs: Recurrence Relations and Closed Formulas. Submitted to Journal of Integer Sequences.
%H A381862 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F A381862 a(n) = 10*binomial(n,6) + 3*n*binomial(n-1,4).
%F A381862 a(n) = n*(n-1)*(n-2)*(n-3)*(n-4)*(n+4)/72.
%F A381862 G.f.: 5*x^5*(3 - x)/(1 - x)^7. - _Stefano Spezia_, Mar 09 2025
%e A381862 a(5) = 15 because there are 15 unordered pairs of triangles that share 1 vertex.
%e A381862 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.
%t A381862 a[n_]:=n*(n-1)*(n-2)*(n-3)*(n-4)*(n+4)/72; Array[a,33,5] (* _Stefano Spezia_, Mar 09 2025 *)
%o A381862 (Python)
%o A381862 def A381862(n): return n*(n*(n*(n*(n*(n-6)-5)+90)-176)+96)//72 # _Chai Wah Wu_, Mar 18 2025
%Y A381862 Cf. A381863, A054647.
%K A381862 nonn,easy
%O A381862 5,1
%A A381862 _Julian Allagan_, Mar 08 2025