A238368 Integer area A of triangles having their side lengths in the commutative ring Z[phi] where phi is the golden ratio.

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

Offset: 1

Author

Michel Lagneau, Feb 25 2014

Keywords

• nonn

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 integer-sided 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*(s-a)*(s-b)*(s-c)), 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*phi-1, 2*phi-1),(3,3-phi,2+phi) and (4,2*phi-1, 2*phi-1) 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.

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| A | a | b | c | R |

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| 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 |

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Links

• Eric Weisstein's World of Mathematics, Ring

Crossrefs

Cf. A188158.

Programs

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*(s-a)*(s-b)*(s-c);If[a*b*c!=0&&N[area2]>0&&Abs[N[Sqrt[area2]]-Round[N[Sqrt[area2]]]]