A364311 Lexicographically earliest infinite sequence of nonnegative integers, {a(n)} for n>=0, such that all lines with equations y = a(n)*x + n are in general position.

0, 1, 3, 2, 5, 4, 8, 11, 6, 13, 12, 7, 22, 16, 17, 21, 9, 14, 10, 27, 18, 15, 19, 28, 31, 43, 34, 38, 23, 39, 25, 36, 41, 20, 55, 63, 42, 30, 24, 33, 26, 32, 65, 66, 51, 59, 29, 56, 35, 62, 85, 81, 37, 49, 46, 68, 74, 78, 88, 48, 44, 75, 40, 47, 97, 76, 93, 79, 92, 54, 58, 100, 61, 101, 107, 52

Offset: 0

Author

Luc Rousseau, Sep 22 2023

Keywords

• nonn

Comments

By "in general position" we mean that no two lines are parallel or coincident, and no three are concurrent.

Examples

			Line #0 is y = 0*x + 0, as no constraint other than the default one (lexicographically earliest) applies yet, so a(0)=0;
Line #1 cannot be y = 0*x + 1, as the reuse of the slope of a previously defined line implies parallelism; but it can be y = 1*x + 1, so a(1)=1;
Line #2 cannot be y = 2*x + 2, as lines #0, #1 and #2 would be concurrent at (-1,0); but Line #2 can be y = 3*x + 2, so a(2)=3; ...
		

Links

• Luc Rousseau, Illustration, the first five lines

Crossrefs

Cf. A000217 (number of intersection points to avoid after line #n is created).

Programs

PARI

g(n,an,p)=my(b=1);for(i=1,#p,if(p[i][2]==an*p[i][1]+n,b=0;break));b
c(n,s,p)=my(an=0);while((setsearch(s,an,0)!=0)||!g(n,an,p),an++);an
w(n,a,an,p)=for(m=0,#a-1,my(am=a[1+m]);listput(p,[(m-n)/(an-am),(am*n-an*m)/(am-an)]));p
list_a(N)=my(a=List(),s=List(),p=List(),n=0,an);while(n<=N,an=c(n,s,p);p=w(n,a,an,p);listput(a,an);listinsert(s,an,setsearch(s,an,1));n++);Vec(a)
list_a(75)