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A371973 a(n) is the number of distinct areas > 0 of triangles with integer sides and perimeter n.

%I A371973 #25 Aug 02 2025 09:59:15
%S A371973 1,0,1,1,2,1,3,2,4,3,5,4,7,5,8,7,10,8,12,10,14,12,16,13,19,14,21,19,
%T A371973 23,20,27,23,30,27,32,29,35,32,39,34,44,39,48,43,52,47,55,51,60,53,63,
%U A371973 59,69,58,74,67,78,73,84,75,90,81,92,88,101,91,108,93,112,106
%N A371973 a(n) is the number of distinct areas > 0 of triangles with integer sides and perimeter n.
%H A371973 Hugo Pfoertner, <a href="/A371973/b371973.txt">Table of n, a(n) for n = 3..10000</a>
%F A371973 a(n) = |{A135622(k) : A070083(k) = n}| = |{A135622(k) : A026810(n) < k <= A026810(n+1)}|. - _Peter Munn_, Jul 29 2025
%o A371973 (PARI) A2(a,b,c) = {my (s=(a+b+c)/2); s*(s-a)*(s-b)*(s-c)};
%o A371973 a371973(n) = {my (A=List()); forpart (v=n, listput(A, A2(v[1],v[2],v[3])), [1,(n-1)\2], [3,3]); #Set(A)};
%o A371973 (Python) def A371973(n): return len(set((2*(b+c)-n)*(n-2*b)*(n-2*c) for c in range((n+2)//3, (n+1)//2) for b in range((n-c+1)//2, c+1))) # _David Radcliffe_, Aug 01 2025
%Y A371973 Cf. A005044, A007237, A024153, A317182, A371070.
%Y A371973 See the formula section for the relationships with A026810, A070083, A135622 (which has many crossrefs related to areas of triangles).
%K A371973 nonn
%O A371973 3,5
%A A371973 _Hugo Pfoertner_, Apr 16 2024
%E A371973 b-file corrected by _David Radcliffe_, Aug 01 2025