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

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, 23, 20, 27, 23, 30, 27, 32, 29, 35, 32, 39, 34, 44, 39, 48, 43, 52, 47, 55, 51, 60, 53, 63, 59, 69, 58, 74, 67, 78, 73, 84, 75, 90, 81, 92, 88, 101, 91, 108, 93, 112, 106

Offset: 3

Author

Hugo Pfoertner, Apr 16 2024

Keywords

• nonn

Links

• Hugo Pfoertner, Table of n, a(n) for n = 3..10000

Crossrefs

Cf. A005044, A007237, A024153, A317182, A371070.

See the formula section for the relationships with A026810, A070083, A135622 (which has many crossrefs related to areas of triangles).

Programs

PARI

A2(a,b,c) = {my (s=(a+b+c)/2); s*(s-a)*(s-b)*(s-c)};
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)};

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

Formula

a(n) = |{A135622(k) : A070083(k) = n}| = |{A135622(k) : A026810(n) < k <= A026810(n+1)}|. - Peter Munn, Jul 29 2025

Extensions

b-file corrected by David Radcliffe, Aug 01 2025