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.

How many different ways can n cents be represented with 1, 5, 10, and 25 cent coins?

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).

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 7. Pascal's Triangle (A007318).

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).

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.

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.

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.

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:**

Weisstein, E W. Pascal's Triangle

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.

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.