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A251607
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Number of non-congruent integer triangles ABC with largest side BC of length n whose opposite vertex A lies inside the closed boundary formed by BC and a locus of points whose real distance from B is x, from C is y and x^3 + y^3 = n^3.
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1
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0, 0, 1, 2, 3, 5, 8, 10, 14, 16, 20, 24, 29, 35, 39, 45, 51, 58, 66, 71, 80, 87, 97, 104, 115, 124, 133, 145, 154, 166, 178, 189, 202, 213, 228, 241, 254, 269, 283, 297, 315, 329, 346, 361, 380, 396, 413, 431, 450, 470, 487, 508, 528, 548, 569, 588, 613, 632, 655, 677, 701
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OFFSET
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1,4
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COMMENTS
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Acute integer triangles ABC with longest side BC of length n (A247588) are segregated from obtuse or right integer triangles with the same longest side BC (A236384) by the closed boundary formed by a semicircle and BC as its diameter. The right integer triangles will lie on this boundary and the obtuse integer triangles within this boundary. Define a closed boundary S(q) that is formed by BC and a locus of points whose real distance from B is x, from C is y and x^q + y^q = n^q for integer q > 0. Then S(2) is that closed boundary formed by a semicircle with BC as diameter. Euler proved that there are no integer triangles that lie on S(3) and Wiles for all S(q) where q > 2. This sequence identifies all integer triangles with longest side BC of length n that lie inside S(3).
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LINKS
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EXAMPLE
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a(5)=3 as there are 3 non-congruent integer triangles with base length of 5 whose apex lies inside S(3). The integer triples are (3,3,5), (2,4,5), (3,4,5). The other triangles from the complete set of non-congruent integer triangles with longest side length 5 (A002620(5+1)) are (4,4,5), (1,5,5), (2,5,5), (3,5,5), (4,5,5), (5,5,5) and lie outside the closed boundary.
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MATHEMATICA
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sumtriangles[c_] := (n = 0; Do[If[a^3+b^3<c^3&&a+b>c, n++], {b, 1, c}, {a, 1, b}]; n); Table[sumtriangles[m], {m, 1, 200}]
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PROG
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(GeoGebra)
n = Slider(1, 20, 1);
L = Flatten(Sequence(Sequence(((a^2+n^2-(n-a+k)^2)/(2n), ((a+(n-a+k)+n)(a+(n-a+k)-n)(a-(n-a+k)+n)(-a+(n-a+k)+n))^(1/2)/(2n)), a, k, (n+k)/2), k, 1, n));
p = 3;
C = Curve((a^2+n^2-(n^p-a^p)^(2/p))/(2n), ((a+(n^p-a^p)^(1/p)+n)(a+(n^p-a^p)^(1/p)-n)(a-(n^p-a^p)^(1/p)+n)(-a+(n^p-a^p)^(1/p)+n))^(1/2)/(2n), a, 0, n);
a_n = CountIf((x(A)^2+y(A)^2)^(p/2)+((n-x(A))^2+y(A)^2)^(p/2)<n^p, A, L);
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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