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

1,2

Links

Table of n, a(n) for n=1..49.

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

Example

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

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}]

Crossrefs

Cf. A331250.

Sequence in context: A077631 A350878 A354246 * A025223 A348919 A177787

Adjacent sequences: A381096 A381097 A381098 * A381101 A381102 A381104

Keyword

easy,nonn,new

Author

Vladimir Reshetnikov, Feb 13 2025

Status

approved