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).

24, 24, 48, 45, 45, 63, 48, 72, 63, 72, 96, 90, 105, 90, 120, 126, 96, 120, 105, 144, 120, 144, 135, 135, 165, 120, 195, 126, 168, 189, 144, 144, 192, 180, 168, 210, 180, 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

Offset: 1

Author

Felix Huber, Jul 07 2024

Keywords

• nonn

Comments

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.

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.

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.

Examples

			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.
		

Crossrefs

Cf. A373995 (values x1), A373997 (maximum values for k), A364384, A364385.

Programs

Maple

A373996:=proc(s)
  local x_1,x_2,x_3,x_4,L;
  L:=[];
  for x_1 from 1 to floor((s-1)/2) do
    x_2:=s-x_1;
    x_3:=(x_1+x_2+sqrt(x_1^2+x_2^2-x_1*x_2))/3;
    x_4:=(x_1+x_2-sqrt(x_1^2+x_2^2-x_1*x_2))/3;
    if x_3=floor(x_3) and x_4=floor(x_4) then
      L:=[op(L),x_2];
    fi;
  od;
  return op(L);
end proc;
seq(A373996(s),s=3..414);

Formula

x-coordinate of the 1. extreme point: x3 = (x1 + x2 + sqrt(x1^2 + x2^2 - x1*x2))/3.

x-coordinate of the 2. extreme point: x4 = (x1 + x2 - sqrt(x1^2 + x2^2 - x1*x2))/3.

x-coordinate of the inflection point: x5 = (x1 + x2)/3 = (x3 + x4)/2.

k = GCD(f(x3), f(x4), f(x5)).

Extensions

Data corrected by Felix Huber, Aug 18 2024