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

1,1

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 - 2*x*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.

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

Table of n, a(n) for n=1..67.

Wikipedia, Steiner inellipse.

Wikipedia, Marden's theorem.

Example

(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.

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:

Crossrefs

Sequence in context: A248644 A242197 A283051 * A376377 A244671 A391349

Adjacent sequences: A383106 A383107 A383108 * A383110 A383111 A383112

Keyword

nonn

Author

Michel Lagneau, Apr 16 2025

Status

approved