On To The Next…

December 30, 2010 at 1:31 am 1 comment

Another year almost over, the next one about to begin. Which makes me think of sequences…

Math jokes make all my Cauchy sequences converge.

And here’s a Fox Trot cartoon with sequences:

You undoubtedly can identify the first sequence: 1, 1, 2, 3, 5, 8, 13, …
Of course, it’s the Fibonacci sequence.

But do you recognize the sequence from the last panel? It begins 3, 0, 2, 3, 2, 5, …

If not, here’s your first question:

What’s the next term in that sequence?

And your second question:

What is the general formula for the terms in that sequence? (A recursive formula is completely acceptable. The explicit formula is quite a beast.)

Like the Fibonacci sequence, this sequence is defined by a recurrence relation. In particular,

P(0) = 3, P(1) = 0, P(2) = 2, and P(n) = P(n – 2) + P(n – 3)

This sequence has an amazing property: For any natural number n, if n is prime, then n | P(n). No, really. You can check for yourself. P(3), P(5), and P(7) are trivial, since P(3) = 3, P(5) = 5, and P(7) = 7. But…

P(11) = 22, and 11|22

P(13) = 39, and 13|39

P(17) = 119, and 17|119

P(19) = 209, and 19|209

P(23) = 644, and 23|644

Also like the Fibonacci sequence, the ratio of consecutive numbers in this sequence have a constant ratio. As we all know, the ratio of consecutive Fibonacci numbers is approximately 1.618034, better known as the golden ratio. For the Perrin sequence, the ratio of consecutive numbers is approximately 1.324718, known as the plastic constant.

Cool stuff.

Here are a couple other sequences for you to ponder as you prepare for the new year. Can you determine the next term?

O, T, T, F, F, S, S, E, …

3, 3, 5, 4, 4, 3, 5, 5, …

6, 14, 24, 36, 50, …

Entry filed under: Uncategorized. Tags: , , , .

Permutations Welcome to 2011

1 Comment Add your own

  • 1. xander  |  December 30, 2010 at 1:58 am

    O, T, T, F, F, S, S, E: The next terms are N, T, E. These are the initials of the counting numbers. (O)ne, (T)wo, (T)hree, and so on.

    3, 3, 5, 4, 4, 3, 5, 5: The next terms are 4, 3, 6. S(n) = the number of letters in the word for the number n. One has three letters, two has three letters, &c.

    6, 14, 24, 36, 50: I think that the next terms are 66, 94, 114. a_0 = 0, a_{n+1} = a_{n} + 2(n+3). I think. Maybe. It has been a while since my theory of positive integers, so I don’t think that I am going to try to work out an explicit formula right now, though I don’t think that it should be that difficult. Call me lazy.😉

    Reply

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

Trackback this post  |  Subscribe to the comments via RSS Feed


About MJ4MF

The Math Jokes 4 Mathy Folks blog is an online extension to the book Math Jokes 4 Mathy Folks. The blog contains jokes submitted by readers, new jokes discovered by the author, details about speaking appearances and workshops, and other random bits of information that might be interesting to the strange folks who like math jokes.

MJ4MF (offline version)

Math Jokes 4 Mathy Folks is available from Amazon, Borders, Barnes & Noble, NCTM, Robert D. Reed Publishers, and other purveyors of exceptional literature.

Past Posts

December 2010
M T W T F S S
« Nov   Jan »
 12345
6789101112
13141516171819
20212223242526
2728293031  

Enter your email address to subscribe to the MJ4MF blog and receive new posts via email.

Join 232 other followers

Visitor Locations

free counters

%d bloggers like this: