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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.

%I A383109 #25 Apr 25 2025 23:39:55
%S A383109 3,5,6,10,9,15,13,15,12,20,15,25,18,30,15,37,15,41,17,39,21,35,26,30,
%T A383109 24,40,27,45,25,51,30,50,39,45,33,55,36,60,29,75,30,74,39,65,51,53,30,
%U A383109 82,34,78,42,70,52,60,45,75,39,85,48,80,51,85,65,75,54,90,61
%N 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.
%C A383109 With z1=0, z2 = x+i*y, z3=x-i*y and P(z)=(z-z1)*(z-z2)*(z-z3) we obtain:
%C A383109 (1) P(z) = z^3 -2x*z^2 + (x^2+y^2)*z.
%C A383109 (2) P’(z)=3*z^2-4*x*z+ x^2+y^2.
%C A383109 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.
%D A383109 Beniamin Bogosel, A Geometric Proof of the Siebeck-Marden Theorem, Amer. Math. Monthly, vol. 125, no 4, 2017, p. 459-463.
%D A383109 A. Eydelzon, On a New Property of the Steiner Inellipse, Amer. Math. Monthly, vol. 127, no 10, 2020, p. 933-935.
%H A383109 Wikipedia, <a href="https://en.wikipedia.org/wiki/Inellipse#Steiner_inellipse">Steiner inellipse</a>
%H A383109 Wikipedia, <a href="https://en.wikipedia.org/wiki/Marden%27s_theorem">Marden's theorem</a>
%e A383109 (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.
%e A383109 (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.
%p A383109 nn:=200:
%p A383109 for x from 1 to nn do:
%p A383109 for y from 1 to nn while(x^2>3*y^2) do:
%p A383109  u:=sqrt(x^2+y^2):v:=2*b:s:=sqrt(x^2+y^2)+y:
%p A383109   z1:=(2*x-sqrt(x^2-3*y^2))/3:z2:=(2*x+sqrt(x^2-3*y^2))/3:
%p A383109   if z1=floor(z1) and z2=floor(z2) then printf(`%d, `,z1):
%p A383109    printf(`%d, `,z2):
%p A383109   else fi:
%p A383109   od:
%p A383109 od:
%K A383109 nonn
%O A383109 1,1
%A A383109 _Michel Lagneau_, Apr 16 2025