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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.

%I A381100 #9 Feb 14 2025 18:59:13
%S A381100 1,2,5,10,18,29,44,62,82,109,141,180,226,279,339,403,475,557,651,755,
%T A381100 870,993,1125,1269,1425,1595,1780,1976,2188,2417,2652,2905,3173,3461,
%U A381100 3769,4090,4436,4788,5161,5558,5968,6405,6857,7340,7840,8355,8893,9463,10048
%N 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.
%H A381100 K. A. Post, <a href="https://doi.org/10.1007/BF01667408">Triangle in a triangle: On a problem of Steinhaus</a>. Geom Dedicata 45, 115-120 (1993).
%e A381100 For n = 2, triangles (1, 1, 1) and (2, 2, 2) can fit inside (2, 2, 2), so a(2) = 2.
%t A381100 ClearAll[checkOnce, triangleInTriangleQ, a];
%t A381100 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];
%t A381100 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}]}]];
%t A381100 a[n_] := Total[Flatten[Table[Boole[triangleInTriangleQ[n, n, n, p, q, r]], {p, n}, {q, p}, {r, p - q + 1, q}]]];
%t A381100 Table[a[n], {n, 1, 49}]
%Y A381100 Cf. A331250.
%K A381100 easy,nonn
%O A381100 1,2
%A A381100 _Vladimir Reshetnikov_, Feb 13 2025