cp's OEIS Frontend

A231275 Integer areas of the outer vecten triangles of integer-sided triangles.

%I A231275 #22 Feb 16 2025 08:33:20
%S A231275 49,91,105,196,289,301,364,379,420,441,459,505,529,609,631,697,751,
%T A231275 784,799,819,889,897,945,961,991,1009,1156,1171,1204,1219,1225,1351,
%U A231275 1395,1401,1456,1516,1521,1611,1680,1681,1689,1759,1764,1836,1849,1939,2020,2095
%N A231275 Integer areas of the outer vecten triangles of integer-sided triangles.
%C A231275 Consider the external erection of three squares on the sides of a triangle ABC. These centers form a triangle IJK. The vertices of this triangle satisfy a number of remarkable properties.
%C A231275 The area of the outer Vecten triangle is A' = A + (a^2 + b^2 + c^2)/8, where A is the area of the reference triangle.
%C A231275 Its side lengths are
%C A231275   a' = sqrt((b^2 + c^2 + 4*A)/2),
%C A231275   b' = sqrt((a^2 + c^2 + 4*A)/2),
%C A231275   c' = sqrt((a^2 + b^2 + 4*A)/2).
%C A231275 The circumcircle of the outer Vecten circle is the outer vecten circle.
%C A231275 Properties of this sequence:
%C A231275 The primitive triangles are 49, 91, 105, 289, 301, ...
%C A231275 The nonprimitive triangles of areas 4*a(n), 9*a(n), ..., p^2*a(n), ... are in the sequence.
%C A231275 It appears that if the triangles are isosceles, one of the sides of the outer vecten triangles is an integer (see the table below).
%C A231275 The following table gives the first values (A, A', a, b, c, a', b', c') where A is the area of the initial triangles, A' is the area of the outer vecten triangles, a, b, c are the integer sides of the initial triangles, and a', b', c' are the sides of the outer vecten triangles.
%C A231275 -----------------------------------------------------------------------
%C A231275 |   A' | A   |  a  |   b |  c   |    a'      |  b'       |  c'        |
%C A231275 -----------------------------------------------------------------------
%C A231275 |   49 |  24  |  6 |   8 | 10  |  sqrt(130) | 2*sqrt(29) | 7*sqrt(2)  |
%C A231275 |   91 |  48  | 10 |  10 | 12  |  sqrt(218) |  sqrt(218) |   14       |
%C A231275 |  105 |  48  | 10 |  10 | 16  |  sqrt(274) |  sqrt(274) |   14       |
%C A231275 |  196 |  96  | 12 |  16 | 20  | 2*sqrt(130)|  4*sqrt(29)| 14*sqrt(2) |
%C A231275 |  289 | 120  | 10 |  24 | 26  |  sqrt(866) | 2*sqrt(157)| 17*sqrt(2) |
%C A231275 |  301 |  96  |  8 |  26 | 30  |14*sqrt(5)  |  sqrt(674) | sqrt(562)  |
%C A231275 |  364 | 192  | 20 |  20 | 24  |2*sqrt(218) | 2*sqrt(218)| 28         |
%C A231275 |  379 | 144  | 18 |  20 | 34  |sqrt(1066)  | 2*sqrt(257)| 5*sqrt(26) |
%C A231275 |  420 |  192 | 20 |  20 | 32  |2*sqrt(274) | 2*sqrt(274)| 28         |
%C A231275 |  441 | 216  | 18 |  24 | 30  |3*sqrt(130) | 6*sqrt(29) | 21*sqrt(2) |
%C A231275 |  459 | 240  | 20 |  26 | 26  |    34      | sqrt(1018) | sqrt(1018) |
%C A231275 |  505 | 168  | 14 |  30 | 40  |sqrt(1586)  | sqrt(1586) | 2*sqrt(221)|
%C A231275 ....................................................................
%D A231275 H. S. M. Coxeter and S. L. Greitzer, Points and Lines Connected with a Triangle, Ch. 1 in Geometry Revisited, Washington, DC, Math. Assoc. Amer., pp. 1-26 and 96-97, 1967.
%H A231275 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OuterVectenTriangle.html">Outer Vecten Triangle</a>
%e A231275 49 is in the sequence. We use two ways:
%e A231275 First way: with the triangle (6, 8, 10) the formula A' = A + (a^2 + b^2 + c^2)/8 gives directly the result: A' = 24 + (6^2 + 8^2 + 10^2)/8 = 24 + 25 = 49 where the area A = 24 is obtained by Heron's formula A = sqrt(s*(s-a)*(s-b)*(s-c)) = sqrt(12*(12-6)*(12-8)*(12-10)) = sqrt(576) = 24, where s is the semiperimeter.
%e A231275 Second way: by calculation of the sides a', b', c' and by use of Heron's formula.
%e A231275   a' = sqrt((b^2 + c^2 + 4*A)/2) = sqrt((8^2 + 10^2 + 4*24)/2) = sqrt(130);
%e A231275   b' = sqrt((a^2 + c^2 + 4*A)/2) = sqrt((6^2 + 10^2 + 4*24)/2) = 2*sqrt(29);
%e A231275   c' = sqrt((a^2 + b^2 + 4*A)/2) = sqrt((6^2 + 8^2 + 4*24)/2) = 7*sqrt(2).
%e A231275 Now we use Heron's formula with (a',b',c').
%e A231275 We find
%e A231275   A'=sqrt(s1*(s1-a')*(s1-b')*(s1-c')) with:
%e A231275   s1 = (a'+b'+c')/2 = (sqrt(130) + 2*sqrt(29) + 7*sqrt(2))/2;
%e A231275 We find A'= 49.
%t A231275 nn=500; lst={}; Do[s =(a + b + c)/2; If[IntegerQ[s], area2 = s (s-a)(s-b) (s-c); If[area2 > 0 && IntegerQ[Sqrt[area2] + (a^2 + b^2 + c^2)/8], AppendTo[lst,  Sqrt[area2] + (a^2 + b^2 + c^2)/8]]], {a,nn}, {b,a}, {c,b}]; Union[lst]
%Y A231275 Cf. A188158, A231276.
%K A231275 nonn
%O A231275 1,1
%A A231275 _Michel Lagneau_, Nov 06 2013