This is in co-ordinate system

\[Alpha] {[0, 0}, {1, 0}} \[CirclePlus] \[Beta] {{0, 0}, {1/2, 
Srqt[3]/2}}, \[Alpha] >= 0, \[Beta] >= 0, 0 <= \[Alpha] + \[Beta] \<= n;

only equilateral triangles are considered, so one computes the number of all equilateral triangles directly:

In[44]:= (* number of vertices *)
Clear[v]
RSolve[v[n + 1] - v[n] - n == 2, v[n], n]
Out[45]= {{v[n] -> -(1/2) (-2 - n) (1 + n) + C[1]}}

In[46]:= Clear[v]
v[n_Integer?NonNegative] := (n + 1) (n + 2)/2

In[48]:= v[83]
Out[48]= 3570

In[57]:= (* number of length 1 edges *)
Clear[ed]
RSolve[ed[n + 1] - ed[n] == 3 (n + 1), ed[n], n]
Out[58]= {{ed[n] -> -(3/2) (-1 - n) n + C[1]}}

In[59]:= Clear[ed]
ed[n_Integer?NonNegative] := 3 n (n + 1)/2

In[64]:= ed[4]
Out[64]= 30

In[85]:= (* number of edge length 1 equilateral triangles *)
Clear[t]
t[n_Integer?NonNegative] := n^2


In[87]:= (* number of all equilateral triangles *)
Clear[tt]
tt[n_Integer?NonNegative] := 
 t[n] + Sum[v[n - m], {m, 2, n}] + Sum[v[n - 2 m], {m, 2, Floor[n/2]}]


In[89]:= tt /@ Range[0, 7]
Out[89]= {0, 1, 5, 13, 27, 48, 78, 118}

The border of points in set is itself an equilateral triangle. So one should count: the equilateral triangles of edge length 1 by their leftmost corner

In[28]:= FindInstance[t[p] == v[p - 1] + v[p - 2], p, Integers, 3]
Out[28]= {{p -> 226}, {p -> 31}, {p -> 675}} 

this is an identity and of course

• v[p-1] is the number of upright ( $\Delta$) triangles and
• v[p-2] is the number of downright ( $\nabla$) triangles.

Then here

(* number of all equilateral triangles *)
Clear[tt]
tt[n_Integer?NonNegative] := 
 t[n] + Sum[v[n - m], {m, 2, n}] + Sum[v[n - 2 m], {m, 2, Floor[n/2]}]

gives the second summand the number of upright triangles with edge length m and the third summand is the number of downright triangles with edge length m.

Run

In[11]:= FindLinearRecurrence[tt /@ Range[0, 44]]
Out[11]= {3, -2, -2, 3, -1}

In[24]:= (* shift by 1 *)
LinearRecurrence[{3, -2, -2, 3, -1}, {0, 1, 5, 13, 27, 48}, #] & /@ {{0, 4}, {44, 45}}
Out[24]= {{0, 0, 1, 5, 13}, {21043, 22517}}

In[33]:= RSolve[a[n + 5] == 
                Reverse[{3, -2, -2, 3, -1}] . (a[n + #] & /@ Range[0, 4]), a[n], n]
Out[33]= {{a[n] -> (-1)^n C[1] + C[2] + n C[3] + n^2 C[4] + n^3 C[5]}}

to catch up with A002717 - OEIS of The On-Line Encyclopedia of Integer Sequences.