A383109 Consider the isosceles triangle whose vertices are the Gaussian integers z1=0, z2 = x+i*y, z3=x-i*y. The sequence lists the pairs of positive integer foci (f_i, f_j), f_i < f_j of Steiner inellipse for some z2, z3.

3, 5, 6, 10, 9, 15, 13, 15, 12, 20, 15, 25, 18, 30, 15, 37, 15, 41, 17, 39, 21, 35, 26, 30, 24, 40, 27, 45, 25, 51, 30, 50, 39, 45, 33, 55, 36, 60, 29, 75, 30, 74, 39, 65, 51, 53, 30, 82, 34, 78, 42, 70, 52, 60, 45, 75, 39, 85, 48, 80, 51, 85, 65, 75, 54, 90, 61

Offset: 1

Author

Michel Lagneau, Apr 16 2025

Keywords

• nonn

Comments

With z1=0, z2 = x+i*y, z3=x-i*y and P(z)=(z-z1)*(z-z2)*(z-z3) we obtain:

(1) P(z) = z^3 -2x*z^2 + (x^2+y^2)*z.

(2) P’(z)=3*z^2-4*x*z+ x^2+y^2.

The two zeros of P’(z) are: (2*x-sqrt(x^2-3*y^2))/3 and (2*x+sqrt(x^2-3*y^2))/3.

Examples

			(a(1),a(2)) = (3,5) because from (2) with (x,y) = (6,3), the two zeros of P’(z) are: (2*6-sqrt(6^2-3*3^2))/3 = (12 - sqrt(9))/3 = 3 and (2*6+sqrt(6^2-3*3^2))/3 = (12 + sqrt(9))/3 = 5. The two foci are integers.
(a(7),a(8)) = (13,15) because from (2) with (x,y) = (21,12), the two zeros of P’(z) are (2*21-sqrt(21^2-3*12^2))/3 = (42 - sqrt(9))/3 = 13 and (2*21+sqrt(21^2-3*12^2))/3 = (12 + sqrt(9))/3 = 15. The two foci are integers.
		

References

• Beniamin Bogosel, A Geometric Proof of the Siebeck-Marden Theorem, Amer. Math. Monthly, vol. 125, no 4, 2017, p. 459-463.
• A. Eydelzon, On a New Property of the Steiner Inellipse, Amer. Math. Monthly, vol. 127, no 10, 2020, p. 933-935.

Links

• Wikipedia, Steiner inellipse
• Wikipedia, Marden's theorem

Programs

Maple

nn:=200:
for x from 1 to nn do:
for y from 1 to nn while(x^2>3*y^2) do:
 u:=sqrt(x^2+y^2):v:=2*b:s:=sqrt(x^2+y^2)+y:
  z1:=(2*x-sqrt(x^2-3*y^2))/3:z2:=(2*x+sqrt(x^2-3*y^2))/3:
  if z1=floor(z1) and z2=floor(z2) then printf(`%d, `,z1):
   printf(`%d, `,z2):
  else fi:
  od:
od: