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

0,3

Comments

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

Links

Table of n, a(n) for n=0..75.

Luc Rousseau, Illustration, the first five lines

Example

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; ...

Prog

(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)

Crossrefs

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

Sequence in context: A339371 A258259 A240182 * A115297 A275900 A375498

Adjacent sequences: A364308 A364309 A364310 * A364312 A364313 A364314

Keyword

nonn

Author

Luc Rousseau, Sep 22 2023

Status

approved