cp's OEIS Frontend

A373995 Zeros x1 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 A373995 #15 Aug 18 2024 20:24:58
%S A373995 9,15,18,21,24,15,30,27,48,45,36,42,33,48,21,30,60,45,72,39,75,54,63,
%T A373995 72,48,99,27,96,63,45,90,105,72,84,105,66,96,42,117,51,81,60,120,33,
%U A373995 96,90,105,144,120,135,57,144,99,168,78,135,75,150,120,108,126,99,144
%N A373995 Zeros x1 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 A373995 The corresponding values x2 are in A373996. The corresponding maximum values for k, for which the y-coordinates of the extreme points and the inflection point are integers, are in A373997.
%C A373995 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 A373995 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 A373995 x-coordinate of the 1. extreme point: x3 = (x1 + x2 + sqrt(x1^2 + x2^2 - x1*x2))/3.
%F A373995 x-coordinate of the 2. extreme point: x4 = (x1 + x2 - sqrt(x1^2 + x2^2 - x1*x2))/3.
%F A373995 x-coordinate of the inflection point: x5 = (x1 + x2)/3 = (x3 + x4)/2.
%F A373995 k = GCD(f(x3), f(x4), f(x5)).
%e A373995 9 is 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.
%e A373995 24 is in the sequence, since the x-cordinates of the extreme points and of the inflection point of f(x) = 1/k*x*(x - 24)*(x - 45) are 10, 36 and 23.
%p A373995 A373995:=proc(s)
%p A373995   local x_1,x_2,x_3,x_4,L;
%p A373995   L:=[];
%p A373995   for x_1 from 1 to floor((s-1)/2) do
%p A373995     x_2:=s-x_1;
%p A373995     x_3:=(x_1+x_2+sqrt(x_1^2+x_2^2-x_1*x_2))/3;
%p A373995     x_4:=(x_1+x_2-sqrt(x_1^2+x_2^2-x_1*x_2))/3;
%p A373995     if x_3=floor(x_3) and x_4=floor(x_4) then
%p A373995       L:=[op(L),x_1];
%p A373995     fi;
%p A373995   od;
%p A373995   return op(L);
%p A373995 end proc;
%p A373995 seq(A373995(s),s=3..414);
%Y A373995 Cf. A373996 (values x2), A373997 (maximum values for k), A364384, A364385.
%K A373995 nonn
%O A373995 1,1
%A A373995 _Felix Huber_, Jun 24 2024
%E A373995 Data corrected by _Felix Huber_, Aug 18 2024