A381100 Number of integer triples i <= j <= k such that a non-degenerate triangle with sides (i, j, k) fits inside an equilateral triangle with sides (n, n, n), possibly touching its boundary from inside.

1, 2, 5, 10, 18, 29, 44, 62, 82, 109, 141, 180, 226, 279, 339, 403, 475, 557, 651, 755, 870, 993, 1125, 1269, 1425, 1595, 1780, 1976, 2188, 2417, 2652, 2905, 3173, 3461, 3769, 4090, 4436, 4788, 5161, 5558, 5968, 6405, 6857, 7340, 7840, 8355, 8893, 9463, 10048

Offset: 1

Author

Vladimir Reshetnikov, Feb 13 2025

Keywords

• easy
• nonn

Examples

			For n = 2, triangles (1, 1, 1) and (2, 2, 2) can fit inside (2, 2, 2), so a(2) = 2.
		

Links

• K. A. Post, Triangle in a triangle: On a problem of Steinhaus. Geom Dedicata 45, 115-120 (1993).

Crossrefs

Cf. A331250.

Programs

Mathematica

ClearAll[checkOnce, triangleInTriangleQ, a];
checkOnce[{a_, b_, c_}, {p_, q_, r_}] := With[{d = (a + b - c) (a - b + c) (-a + b + c) (a + b + c), s = (p + q - r) (p - q + r) (-p + q + r) (p + q + r), u = p^2 + q^2 - r^2, v = p^2 - q^2 + r^2}, p <= a && a^2 s <= d p^2 && u v >= 0 && s (a^2 - b^2 + c^2)^2 <= d (2 a p - u)^2 && s (a^2 + b^2 - c^2)^2 <= d (2 a p - v)^2];
triangleInTriangleQ[a_, b_, c_, p_, q_, r_] := Or @@ Flatten[Table[checkOnce[abc, pqr], {abc, {{a, b, c}, {b, c, a}, {c, a, b}}}, {pqr, Permutations[{p, q, r}]}]];
a[n_] := Total[Flatten[Table[Boole[triangleInTriangleQ[n, n, n, p, q, r]], {p, n}, {q, p}, {r, p - q + 1, q}]]];
Table[a[n], {n, 1, 49}]