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# Pictures from the OEIS

## The (Free) OEIS Store

The On-Line Encyclopedia of Integer Sequences (or OEIS) is a huge collection of number sequences, maintained by The OEIS Foundation.
Many of these sequences have spectacular illustrations, some of which are shown here.
They would look pretty good on T-shirts, tote bags, banners, posters, coffee cups, etc.

but remembering how much we hate journals that hide articles behind paywalls, we decided against it.

However, if you make use of any of these pictures, we hope you will consider making a donation to The OEIS Foundation.

Thumbnail Name,
A-number
(Credits)
Brief
Description
Pictures
Peaceable Queens:
A250000
(Smith-Petrie-Gent)
Place k white queens, k black queens on n X n board so they don't attack each other.
Fig. shows 11X11 board, where k=17 is maximal, illustrating A250000(11) = 17.
Only 13 terms are known.
Peaceable Queens (continued):
A250000
(Michael De Vlieger)
"Peace to the Max" T-shirt showing maximal non-attacking arrangement of 17 black queens and 17 white queens on 11X11 board, illustrating
A250000(11) = 17.
The toothpick sequence:
A139250
(Applegate-Pol-Sloane)
Fig. shows structure after 23 generations, when there are 283 toothpicks, illustrating
A139250(23) = 283.
Toothpick sequence (cont.):
A139250
(Michael De Vlieger)
T-shirt designs from Michael De Vlieger. Fig. on left shows structure after 28 generations, when there are 423 toothpicks, illustrating
A139250(28) = 423.
E-shaped toothpicks:
A161330
(Omar Pol,
D. Applegate)
Fig. shows structure after 32 generations, when there are 1124 E-toothpicks, illustrating
A161330(32) = 1124.
More toothpick structures:
For a very large number of similar structures, both pictures and animations, see David Applegate's Toothpick Movie Page
Fig. shows gullwing toothpick structure after 16 generations, illustrating A187220(16)=32.
For thousands of similar pictures, see the Toothpick Movie Page
Click "pdf" button to save the pictures.
Fredkin's Replicator:
A160239
(N. J. A. Sloane,
Mathematica)
Number of ON cells after n generations in Fredkin's Replicator.
After 15 generations there are 416 ON cells, so
A160239(15) = 416,
Fastest-growing
Odd-Rule CA:
A255462
(N. J. A. Sloane,
Mathematica)
Number of ON cells after n generations in Odd-Rule Cellular Automaton 365.
After 15 generations there are 606 ON cells, so
A255462(15) = 606.
Square grid with no red square:
A227133
(Giovanni Resta)
Maximal number of points in nXn grid such that no 4 form a square. Figure shows 41 red points in 8 X 8 grid such that no 4 red points form a square,
illustrating A227133(8) = 41.
Only 10 terms are known.
A corner design:
A232467
(Craig S. Kaplan)
A (20,2) corner design of C. S. Kaplan (Bridges, 2013), "redrawn from the menu of
Os Tibetanos, a Tibetan restaurant in Lisbon".
Circles in the plane:
A250001
(N. J. A. Sloane)
Number of ways to draw n circles in the plane.
Shows 7 of the 14 ways to draw 3 circles, partly illustrating A250001(3) = 14.
Only 5 terms are known.
Meanders in nXn grid:
A200000
(Jonathan Wild)
There are 42 different meanders in a 5X5 grid, so
A200000(5) = 42.
Coloring empires:
A230628
(Ian Stewart)
Each empire has n countries: how many colors are needed? n=1 is 4-color problem. Figure shows case n=2 which requires 12 colors, illustrating
A230628(2) = 12.
Dissecting
polygon to square:
A110312
(V. Vaishampayan)
Minimal number of pieces for dissecting an n-gon into a square. Conjecturally, 4 pieces are needed for the triangle to square problem,
so A110312(3) = 4 (?).
Kobon triangles:
A006066
Maximal number of triangles formed from n lines drawn in the plane. Figure shows optimal arrangement of 17 lines, giving 85 triangles, illustrating A006066(17) = 85.
Only 9 terms are known.
Packing points in a triangle:
A243487
(Robert Israel)
Largest minimal Manhattan distance for n points in a simplex: Figure shows optimal arrangement of 17 points, illustrating A243487(17) / A243576(17) = 6/13.
Min number of pieces
in dissection of polygon:
A160860
Min number of pieces in convex n-gon with all diagonals drawn.
For heptagon, 47 pieces is minimal: A160860(7) = 47.
Only known for n<9.
Figurate numbers (1):
A169721
(Alice V. Kleeva)
Geometrical design based on the figurate numbers of Pythagoras and on the regular division of the plane by the square grid.
Figurate numbers (2):
A169727
(Alice V. Kleeva)
Geometrical design based on the figurate numbers of Pythagoras and on a regular division of the plane into a grid of hexagons and squares.
Coordination sequence
for 3.3.3.3.6 tiling:
A250120
(Darrah Chavey)
Number of points at distance n
from origin in planar net 3.3.3.3.6.
Isolated semiprimes:
A113688
(Alois Heinz)
Spiral showing all semiprimes ≤ 10000, with isolated semiprimes in red.
Self-generating sequence avoiding arithmetic progression:
A229037
(J. W. Grahl, X. Gregg)
a(n) is as small as possible such that no three terms a(j), a(j+k), a(j+2k) form an arithmetic progression.
Graph of 10000 terms.
Space-filling curve of order 3:
A265671
(Joerg Arndt)
Third stage of a space-filling curve.
This is a single curve
Non-crossing partitions:
A000108
(Tilman Piesk)

Among other things, the Catalan numbers count
non-crossing partitions.
Figure shows the 42
non-crossing partitions on five points: A000108(5) = 42.

Somos-6 surface:
A006722
(Fedorov-Hone)
Projection of 4-dimensional surface defined by
Somos-6 sequence.

## The OEIS Posters

Poster illustrating A259934
and A263267. Subtract the number of divisors from a number in the tree, and you get the number under it. Shows the inter-connectedness of the OEIS.
(Michael De Vlieger and Antti Kartunen)
The New OEIS Poster
Illustrates nine recent sequences, out of a quarter of a million.
(N. J. A. Sloane)
The Blue OEIS Poster
(Lucas Garron)
The Original OEIS Poster
The original poster, created in 2009.
(D. Applegate, N. J. A. Sloane)

## Captions for the Front of the T-Shirt

If you print any of these pictures on the back of a T-shirt,
on the front you could say any of the following:

OEIS.org:
What comes next after 0, 1, 3, 6, 2, 7, 13, 20, 12 ?

OEIS.org:
Identifying number sequences since 1964

The On-Line Encyclopedia of Integer Sequences:
Identifying number sequences since 1964

Copperplate font is our current favorite.

## Sources for these pictures

Suggestions for additional pictures will be welcomed -- send them to Neil Sloane, njasloane@gmail.com

Most of these pictures appear (or will soon appear) in entries in the OEIS, although some have been reformatted.
See the appropriate entries in the OEIS for information about the original sources for the sequences,
as well as for more precise definitions. The A-number of the sequence is given next to the thumbnail picture.
To find out more about the first picture, for example, which is from sequence A250000, go to https://oeis.org/A250000.

Many of the pictures are the combined work of several people —
the author of the work where the sequence first appeared, the person who contributed the sequence to the OEIS,
the person who created the illustration: these may or not be the same person.

Special thanks to:
Joerg Arndt, David Applegate, Johannes Bader, Darrah Chavey, Michael De Vlieger,
Yuri N. Fedorov and Andrew N. W. Hone, Lucas Garron, Jack W. Grahl, Xan Gregg, Brian Hayes,
Alois Heinz, Robert Israel, Craig S. Kaplan, Antti Kartunen, Alice V. Kleeva, Vladimir Letsko,
Tilman Piesk, Omar E. Pol, Giovanni Resta, Barbara M. Smith, Karen E. Petrie and Ian P. Gent,
Ian Stewart, Vinay Vaishampayan, Jonathan Wild,
whose pictures are included here.

This page was created in Nov.-Dec. 2015 by N. J. A. Sloane and Robert Munafo, following a suggestion from L. Edson Jeffery.
Special thanks to Michael De Vlieger, whose design skills greatly exceed our own.

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Last modified December 10 08:16 EST 2018. Contains 318045 sequences. (Running on oeis4.)