OFFSET
3,4
COMMENTS
The number of scalene triangles equals (number of triangles, i.e., binomial(n,3)) - (number of isosceles triangles).
The general formula is readily proved true by counting arguments.
LINKS
Colin Barker, Table of n, a(n) for n = 3..1000
Index entries for linear recurrences with constant coefficients, signature (0,2,2,-1,-4,-1,2,2,0,-1).
FORMULA
a(n) = binomial(n,3) - A320577(n).
a(n) = C(n,3)-n*(n-1)/2 if n mod 6 = 1 or 5; C(n,3)-n*(n-2)/2 if n mod 6 = 2 or 4; C(n,3)-n*(3*n-7)/6 if n mod 6 = 3; C(n,3)-n*(3*n-10)/6 otherwise [C(n,k) denoting binomial coefficients].
G.f.: 2*x^6*(2+x)*(3+x*(2+x))/((x-1)^4*(x+1)^2*(1+x+x^2)^2).
a(n) = 2*a(n-2) + 2*a(n-3) - a(n-4) - 4*a(n-5) - a(n-6) + 2*a(n-7) + 2*a(n-8) - a(n-10) for n>12. - Colin Barker, Jan 08 2020
EXAMPLE
Trivial cases:
a(3)=0 since the only triangle formed by joining vertices is equilateral.
a(4)=a(5)=0 since all such triangles are isosceles.
For higher n, since a triangle is formed by choosing 3 vertices and joining them, there are C(n,3) such triangles. To obtain the number of scalene triangles, subtract the number of isosceles triangles (A320577).
MATHEMATICA
a[n_] := If[Mod[n, 6]==1 || Mod[n, 6]==5, Binomial[n, 3]-Binomial[n, 2], If[Mod[n, 6]==2 || Mod[n, 6]==4, Binomial[n, 3]-n*(n-2)/2,
If[Mod[n, 6]==3, Binomial[n, 3]-n*(3*n-7)/6, Binomial[n, 3]-n*(3*n - 10)/6]]]; Array[a, 20, 3]
LinearRecurrence[{0, 2, 2, -1, -4, -1, 2, 2, 0, -1}, {0, 0, 0, 12, 14, 32, 54, 80, 110, 168}, 50] (* Harvey P. Dale, Aug 20 2021 *)
PROG
(Python)
from sympy import binomial
def a(n):
assert (n>=3), "Sequence a(n) defined for n>=3"
m = n % 6
Cn3 = binomial(n, 3)
if m in [1, 5]: return Cn3 - (n*(n-1))//2
elif m in [2, 4]: return Cn3 - (n*(n-2))//2
elif m==3: return Cn3 - (n*(3*n-7))//6
else: return Cn3 - (n*(3*n-10))//6
print([a(k) for k in range(3, 51)])
(PARI) concat([0, 0, 0], Vec(2*x^6*(2 + x)*(3 + 2*x + x^2) / ((1 - x)^4*(1 + x)^2*(1 + x + x^2)^2) + O(x^60))) \\ Colin Barker, Jan 08 2020
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Adam Vellender, Dec 05 2019
STATUS
approved