cp's OEIS Frontend

A373996 Zeros x2 of polynomial functions f(x) = 1/k*x*(x - x1)*(x - x2), which have three integer zeros 0, x1 and x2 (with 0 < x1 < x2) as well as two extreme points and one inflection point with integer x-coordinates (sorted in ascending order, first by the sum x1 + x2 and then by x1).

%I A373996 #13 Aug 19 2024 11:43:14
%S A373996 24,24,48,45,45,63,48,72,63,72,96,90,105,90,120,126,96,120,105,144,
%T A373996 120,144,135,135,165,120,195,126,168,189,144,144,192,180,168,210,180,
%U A373996 240,165,240,216,252,192,288,231,240,225,189,225,216,297,210,264,195,288,231,315,240,273,288,270,315,270
%N A373996 Zeros x2 of polynomial functions f(x) = 1/k*x*(x - x1)*(x - x2), which have three integer zeros 0, x1 and x2 (with 0 < x1 < x2) as well as two extreme points and one inflection point with integer x-coordinates (sorted in ascending order, first by the sum x1 + x2 and then by x1).
%C A373996 The corresponding values x1 are in A373995. The corresponding maximum values for k, for which the y-coordinates of the extrema and the inflection are integers, are in A373997.
%C A373996 These polynomial functions can be used in math lessons when discussing curves. Zeros, extreme points and inflection points can be determined without unnecessary calculation effort with fractions and roots.
%C A373996 Of course, these functions can be stretched in the y-direction by a factor 1/k without affecting the zeros, the extreme points and the inflection point, or shifted in the x-direction, whereby the zeros, the extreme points and the inflection point are also shifted.
%F A373996 x-coordinate of the 1. extreme point: x3 = (x1 + x2 + sqrt(x1^2 + x2^2 - x1*x2))/3.
%F A373996 x-coordinate of the 2. extreme point: x4 = (x1 + x2 - sqrt(x1^2 + x2^2 - x1*x2))/3.
%F A373996 x-coordinate of the inflection point: x5 = (x1 + x2)/3 = (x3 + x4)/2.
%F A373996 k = GCD(f(x3), f(x4), f(x5)).
%e A373996 24 is twice in the sequence, since the x-cordinates of the extreme points and of the inflection point of f(x) = 1/k*x*(x - 9)*(x - 24) are 4, 18 and 11 and of f(x) = 1/k*x*(x - 15)*(x - 24) are 6, 20 and 13.
%p A373996 A373996:=proc(s)
%p A373996   local x_1,x_2,x_3,x_4,L;
%p A373996   L:=[];
%p A373996   for x_1 from 1 to floor((s-1)/2) do
%p A373996     x_2:=s-x_1;
%p A373996     x_3:=(x_1+x_2+sqrt(x_1^2+x_2^2-x_1*x_2))/3;
%p A373996     x_4:=(x_1+x_2-sqrt(x_1^2+x_2^2-x_1*x_2))/3;
%p A373996     if x_3=floor(x_3) and x_4=floor(x_4) then
%p A373996       L:=[op(L),x_2];
%p A373996     fi;
%p A373996   od;
%p A373996   return op(L);
%p A373996 end proc;
%p A373996 seq(A373996(s),s=3..414);
%Y A373996 Cf. A373995 (values x1), A373997 (maximum values for k), A364384, A364385.
%K A373996 nonn
%O A373996 1,1
%A A373996 _Felix Huber_, Jul 07 2024
%E A373996 Data corrected by _Felix Huber_, Aug 18 2024