

A238368


Integer area A of triangles having their side lengths in the commutative ring Z[phi] where phi is the golden ratio.


1



1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 16, 18, 20, 22, 24, 25, 27, 28, 29, 30, 32, 33, 34, 35, 36, 38, 40, 42, 44, 45, 48, 49, 50, 54, 55, 56, 58, 60, 62, 63, 64, 66, 68, 70, 72, 75, 76, 77, 78, 80, 81, 84, 88, 90, 95, 96, 98, 99, 100, 108, 110, 112, 114, 116
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OFFSET

1,2


COMMENTS

Generalized integer areas triangles in the ring Z[phi] = {a + b*phi a,b in Z}. Z[phi] is a ring because if x = a + b*phi and y = c + d*phi are in the ring, the sum x+y = a+c + (b+d)*phi is in the ring, and the product x*y = (a*c + b*d) + (a*d + b*c + b*d)*phi is also in the ring.
This sequence is tested with a and b in the range [40, ..., +40]. For the values of areas > 150 it is necessary to expand the range of variation, but nevertheless the calculations become very long.
The sequence A188158 is included in this sequence. The numbers 5*a(n) are in the sequence because if the integer area of the integersided triangle (a, b, c) is A, the area of the triangle of sides (a*sqrt(5), b*sqrt(5), c*sqrt(5)) is 5*A, where sqrt(5)= 1 + 2*phi.
The primitive areas are p = 1, 2, 3, 6, 7, 11, 22, ... and the areas p^2*a(n) are also in the sequence.
The area A of a triangle whose sides have lengths a, b, and c is given by Heron's formula: A = sqrt(s*(sa)*(sb)*(sc)), where s = (a+b+c)/2.
For the same area, the number of triangles is not unique, for example the area of the triangles (1,5,2+4*phi), (2,2*phi1, 2*phi1),(3,3phi,2+phi) and (4,2*phi1, 2*phi1) is A = 2.
It is possible to obtain rational values and also values in the ring Z[phi] for the circumradius (see the table below).
The following table gives the first values (A, a, b, c, R) where A is the integer area, a,b,c are the sides in Z[phi] and R = a*b*c/(4*A) are the values of circumradius.

 A  a  b  c  R 

 1  1  2  1 + 2*phi  sqrt(5)/2 
 2  1  5  2 + 4*phi  5*sqrt(5)/4 
 3  3  1 + 2*phi  2 + 4*phi  5/2 
 4  2  4  2 + 4*phi  sqrt(5) = 1 + 2*phi 
 5  2  13  5 + 10*phi  13*sqrt(5)/2 
 6  3  4  5  5/2 
 7  7  2 + 4*phi  5 + 10*phi  25/2 
 8  5  13  8 + 16*phi  65*sqrt(5)/4 
 10  5  5  2 + 4*phi  5*sqrt(5)/4 
 11  2  11  5 + 10*phi  5*sqrt(5)/2 
 12  4  10  6 + 12*phi  5*sqrt(5) = 5 + 10*phi
 15  5  10  3 + 6*phi  5*sqrt(5)/2 
 16  4  8  4 + 8*phi  2*sqrt(5) = 2 + 4*phi 
 18  3  15  6 + 12*phi  15*sqrt(5)/4 
 20  10  2 + 4*phi  4 + 8*phi  15*sqrt(5)/4 



LINKS

Eric Weisstein's World of Mathematics, Ring


MATHEMATICA

err=1/10^10; nn=40; q=(Sqrt[5]+1)/2; lst={}; lst1={}; Do[If[u+q*v>0, lst=Union[lst, {u+q*v}]], {u, nn, nn, 1}, {v, nn, nn, 1}]; n1=Length[lst]; Do[a=Part[lst, i]; b=Part[lst, j]; c=Part[lst, k]; s=(a+b+c)/2; area2=s*(sa)*(sb)*(sc); If[a*b*c!=0&&N[area2]>0&&Abs[N[Sqrt[area2]]Round[N[Sqrt[area2]]]]<err, AppendTo[lst1, Round[Sqrt[N[area2]]]]], {i, 1, n1}, {j, i, n1}, {k, j, n1}]; Union[lst1]


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



