Math Games: Sequence Pictures

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 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 (A 001203), 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.

(*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.*)