cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A338041 Draw n rays from each of two distinct points in the plane; a(n) is the number of regions thus created. See Comments for details.

Original entry on oeis.org

1, 2, 7, 6, 15, 12, 25, 20, 37, 30, 51, 42, 67, 56, 85, 72, 105, 90, 127, 110, 151, 132, 177, 156, 205, 182, 235, 210, 267, 240, 301, 272, 337, 306, 375, 342, 415, 380, 457, 420, 501, 462, 547, 506, 595, 552, 645, 600, 697, 650, 751, 702, 807, 756, 865, 812, 925
Offset: 1

Views

Author

Lars Blomberg, Oct 08 2020

Keywords

Comments

The rays are evenly spaced around each point. The first ray of one point goes opposite to the direction to the other point. Should a ray hit the other point it terminates there, that is, it is converted to a line segment.
To produce the illustrations below, all pairwise intersections between the rays is calculated and the maximum distance to the center, incremented by 20%, is taken as radius of a circle. Then all intersections between the rays and the circle defines a polygon which is used as limit.

Examples

			For n=1:    <-----x     x----->   so a(1)=1.
For n=2:    <-----x<--->x----->   so a(2)=2.

Links

Crossrefs

Cf. A338042 (vertices), A338043 (edges).

Programs

  • PARI
    a(n)=if(n%2==1,(n^2 + 8*n - 5)/4,(n^2 + 2*n)/4);
vector(200, n, a(n))

Formula

a(n) = (n^2 + 8*n - 5)/4, n odd; (n^2 + 2*n)/4, n even (conjectured).
Conjectured by Stefano Spezia, Oct 08 2020 after Lars Blomberg: (Start)
G.f.: x*(1 + x + 3*x^2 - 3*x^3)/((1 - x)^3*(1 + x)^2).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 5. (End)

A338123 Place three points evenly spaced around a circle, draw n evenly spaced rays from each of the points, a(n) is the number of vertices thus created. See Comments for details.

Original entry on oeis.org

3, 4, 15, 19, 33, 31, 63, 55, 78, 82, 120, 67, 162, 154, 189, 175, 261, 217, 327, 259, 360, 370, 456, 283, 534, 514, 579, 523, 705, 619, 807, 703, 858, 874, 1008, 691, 1122, 1090, 1185, 1111, 1365, 1237, 1503, 1339, 1572, 1594, 1776, 1339, 1926, 1882, 2007, 1891
Offset: 1

Views

Author

Lars Blomberg, Oct 11 2020

Keywords

Comments

The rays are evenly spaced around each point. The first ray from each point goes opposite to the direction to the center of the circle. Should a ray hit another point it is terminated there.
See A338122 for illustrations.

Examples

			For n=1 there are three rays that do not intersect, so a(1)=3.

Links

Crossrefs

Cf. A338042 (two start points), A338122 (regions), A338124 (edges).

Programs

  • PARI
    a(n)=if( \
n%6==5,(3*n^2 + 10*n + 7)/4, \
n%12==1,(3*n^2 + 10*n + 11)/4, \
n%12==2||n%12==10,(3*n^2 + 28)/4, \
n%12==3,(3*n^2 + 4*n + 21)/4, \
n%12==6,(3*n^2 - 10*n + 76)/4, \
n%12==7,(3*n^2 + 10*n + 35)/4, \
n%12==9,(3*n^2 + 4*n + 33)/4, \
n%24==4||n%24==20,(3*n^2 - 12*n + 76)/4, \
n%24==8||n%24==16,(3*n^2 - 12*n + 124)/4, \
n%120==0,(3*n^2 - 40*n - 20)/4, \
n%120==12||n%120==36||n%120==84||n%120==108,(3*n^2 - 40*n + 316)/4, \
n%120==24||n%120==48||n%120==72||n%120==96,(3*n^2 - 40*n + 364)/4, \
n%120==60,(3*n^2 - 40*n - 68)/4, \
-1);
vector(798, n, a(n+2))

Formula

a(n) = 2160-a(n-4)+a(n-12)+a(n-16)+a(n-60)+a(n-64)-a(n-72)-a(n-76), n>78. (conjectured)
From Lars Blomberg, Oct 25 2020: (Start)
Conjectured for 3 <= n <= 800.
Select the row in the table below for which r = n mod m. Then a(n)=(a*n^2 + b*n + c)/d.
+===========================================+
| r | m | a | b | c | d |
+-------------------------------------------+
| 5 | 6 | 3 | 10 | 7 | 4 |
| 1 | 12 | 3 | 10 | 11 | 4 |
| 2, 10 | 12 | 3 | | 28 | 4 |
| 3 | 12 | 3 | 4 | 21 | 4 |
| 6 | 12 | 3 | -10 | 76 | 4 |
| 7 | 12 | 3 | 10 | 35 | 4 |
| 9 | 12 | 3 | 4 | 33 | 4 |
| 4, 20 | 24 | 3 | -12 | 76 | 4 |
| 8, 16 | 24 | 3 | -12 | 124 | 4 |
| 0 | 120 | 3 | -40 | -20 | 4 |
| 12, 36, 84, 108 | 120 | 3 | -40 | 316 | 4 |
| 24, 48, 72, 96 | 120 | 3 | -40 | 364 | 4 |
| 60 | 120 | 3 | -40 | -68 | 4 |
+===========================================+ (End)

A338043 Draw n rays from each of two distinct points in the plane; a(n) is the number of edges thus created. See Comments for details.

Original entry on oeis.org

2, 3, 10, 7, 22, 15, 38, 27, 58, 43, 82, 63, 110, 87, 142, 115, 178, 147, 218, 183, 262, 223, 310, 267, 362, 315, 418, 367, 478, 423, 542, 483, 610, 547, 682, 615, 758, 687, 838, 763, 922, 843, 1010, 927, 1102, 1015, 1198, 1107, 1298, 1203, 1402, 1303, 1510, 1407
Offset: 1

Views

Author

Lars Blomberg, Oct 08 2020

Keywords

Comments

The rays are evenly spaced around each point. The first ray of one point goes in the opposite direction of the other point. Should a ray hit the other point, it terminates there, i.e., it is converted to a line segment.
See A338041 for illustrations.

Examples

			For n=1:    <-----x     x----->   so a(1)=2.
For n=2:    <-----x<--->x----->   so a(2)=3.

Crossrefs

Cf. A338041 (regions), A338042 (vertices).
Cf. also A000096, A000124.

Programs

  • PARI
    a(n)=if(n%2==1,(n^2 + 4*n - 1)/2,(n^2 - 2*n + 6)/2)
vector(200, n, a(n))

Formula

a(n) = (n^2 + 4*n - 1)/2, n odd; (n^2 - 2*n + 6)/2, n even (conjectured).
Conjectured by Stefano Spezia, Oct 08 2020 after Lars Blomberg: (Start)
G.f.: x*(2 + x + 3*x^2 - 5*x^3 + 3*x^4)/((1 - x)^3*(1 + x)^2).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 5. (End)
Hugo Pfoertner, Oct 08 2020: It appears that even terms are 4*A000096+2, odd terms 4*A000124-1.
Showing 1-3 of 3 results.

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