Math Games |
Sequence Pictures |
Ed Pegg Jr., December 8, 2003 |
Mike
Shafer found a term in sequence A000043
last week. When 2 is raised to the power of a number from {2, 3,
5, 7, 13, 17,
19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423,
9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049,
216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377,
6972593,13466917, 20996011}, and one is subtracted, a prime
results. This week, I took an extended look at this and other
sequences in OEIS,
by converting them into pictures. Here's the picture for
Mersenne prime exponents (A000043).
Figure 1. Mersenne Prime
Exponents (
A000043).
I should explain how this picture is made. In binary, 20996011 is
{1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0,
1, 1}. Take a look at the last column in the image, and you'll
see that the binary number has been turned into black and white
squares, reading down. Underneath that, i've placed binary
representations of the natural numbers (1, 2, 3, 4, 5 ...). When
I made this picture, I faintly hoped I might see
some obvious pattern. I didn't.
Here are the whole numbers / natural numbers / positive integers up to
700, in binary columns.

Figure 2. The positive integers (
A000027).
How many different ways can n
cents be represented with 1, 5, 10, and 25 cent coins?
Figure 3. Ways to make change (
A001299).
Here are the triangular numbers, square numbers, and cubic numbers.

Figure 4. Triangular (
A000217),
Square (
A000290),
and Cubic numbers (
A000578).
The Gray Code arranges the numbers so
that only one binary bit changes at a time. It's used in
satellite sensor arrays.

Figure 5. The Gray Code (
A003188).
From the
Small
Groups Library, here is the number of groups of order
n. Note the spikes at powers
of 2.
Figure 6. Groups of order
n. (
A000001).
The numbers in
Pascal's
Triangle, which begin as 1, 1,1, 1,2,1, 1,3,3,1, 1,4,6,4,1
... If you like the picture, you might like to
hear the sequence, at the
Sound of Mathematics
page.

Figure 7. Pascal's Triangle (
A007318).
Other sequence pictures look more
chaotic.
Figure 8. Continued Fraction of Pi (
A001203),
order of SL(2,Z_n) (
A000056),
the Primes (
A000040).
The following lists the number of divisors of n. This is also the number of
Pythagorean triangles with an inscribed circle of radius n.

Figure 9. τ(n): Divisors of
n
(
A000005).
My favorite picture surprised me.
It's the Fibonacci sequence. Until I saw it, I didn't occur to me
that it would have these internal
patterns.

Figure 10. The Fibonacci Sequence (
A000045).
In Stephen Wolfram's book
A New Kind of Science, many
sequence pictures can be found in Chapter 4. The powers of 3/2
makes a fantastic picture. Many more can be seen at the
Color
NKS Images page. As a larger effort,
functions.wolfram.com has
made available thousands of images that can be generated by functions.
For more on sequences, please see N J A Sloane's paper,
My Favorite
Integer Sequences.
References:
Sloane, N J A. Sequences A000001,
A000005,
A000027,
A000040,
A000043,
A000045,
A000056,
A000217,
A000290,
A000578,
A001203,
A001299,
A003188,
A007318
in "The On-Line
Encyclopedia of Integer Sequences."
http://www.research.att.com/~njas/sequences/.
Weisstein, E W. Pascal's
Triangle Eric Weisstein's World of Mathematics.
http://mathworld.wolfram.com/.
Wolfram, S. A New Kind of
Science. Champaign, IL: Wolfram Media, 2002.
Mathematica
Code:
(*Initialization*) RasterGraphics[state_, colors_:2, size_:1] :=
With[{dim =
Reverse[ Dimensions[state]]}, Graphics[ Raster[ Reverse[1 -
state/(colors
- 1)]], AspectRatio -> Automatic, PlotRange -> {{0, dim[[1]]},
{0, dim[[2]]}}, ImageSize -> size*dim + 1]]
(*Figure 10*) With[{seq = Table[Fibonacci[n], {n, 1, 700}]},
Show[ RasterGraphics[ Join[2 Transpose[ Map[ IntegerDigits[#,2,
Ceiling[ Log[2, Max[seq]]]] &,
seq]],Transpose[Table[IntegerDigits[n, 2, 7], {n, 1, Length[seq]}]]],
3, 1]]];
(*For the primes, substitute Primes[n] for Fibonacci[n]. Other
sequences are generated in much the same way, see the below link for
further details.*)
A notebook for all the
images in this column is available at the Mathematica Information Center,
item 5116.
Math Games
archives.
Comments are welcome. Please send comments to Ed Pegg Jr. at
ed@mathpuzzle.com.
Ed Pegg Jr. is the webmaster for mathpuzzle.com.
He works at Wolfram Research, Inc. as the administrator of the
Mathematica
Information Center.