Music of the (binary) trees

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Yesterday I noticed for the first time the "Music" feature of Neil J.A. Sloane's Online Encyclopedia of Integer Sequences, which connects the OEIS's number sequences with Jonathan Middleton's Musical Algorithms site.

So, of course, I immediately listened to sequence A000108, the Catalan numbers, which (among many other things) count the number of ways to arrange n matching pairs of parentheses, or the number of full binary trees with n+1 leaves, or the number of Dyck words of length 2n:

1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, …

Audio clip: Adobe Flash Player (version 9 or above) is required to play this audio clip. Download the latest version here. You also need to have JavaScript enabled in your browser.

This is not the first time that I've posted Dyck music on Language Log — see the discussion and the examples in "Starlings" (4/26/2006). Unlike that earlier example, however, this one has no particular point, other than that OEIS is fun.

If you agree, please contribute to the OEIS foundation. Or just watch the movie:



  1. Emily said,

    July 29, 2011 @ 12:32 pm

    I tried listening to a sequence on the OEIS site and got an error message: "Cannot load data for A-00001." What am I doing wrong?

    Awesome video, though!

  2. Brett said,

    July 29, 2011 @ 1:00 pm

    @Emily: Clicking on one of the suggested sequences doesn't actually work. You apparently have to type the sequence number in manually on the form that it links to.

  3. Neil Sloane said,

    July 29, 2011 @ 3:13 pm

    The best way to listen to a sequence is to enter the name (or A-number, if you know it), e.g. Catalan, Fibonacci, totient or A000108, A005132 – my favorite- or even A123456, worth listening to!). Click "search". Then click the little "listen" link near the top. Neil Sloane

  4. Emily said,

    July 29, 2011 @ 4:48 pm

    @Neil Sloane: Thanks! That works. Now I have a new way to waste time…

  5. slobone said,

    July 29, 2011 @ 7:07 pm

    Where is Conlon Nancarrow when you need him?

  6. AntC said,

    July 31, 2011 @ 12:08 am

    Can anyone help with a distant memory? (I think around late 60's/early 70's BBC Third Programme.) Somebody had recorded a 12-note row with rising and falling arpeggio intervals. Then played it back on two loops of tape with one a very slightly longer repeat cycle. It was mesmerising as the 'tunes' and discords went slowly in and out of synch. (Something like English church bells' change ringing.)

  7. SteveT said,

    July 31, 2011 @ 5:37 pm

    Antc, this?

  8. AntC said,

    August 1, 2011 @ 12:49 am

    Thanks SteveT, but no, not that (which I found irritating rather than mesmeric); nor any of the others that youtube offers in the vicinity.

  9. Dale Gerdemann said,

    August 20, 2011 @ 8:32 pm

    The use of the Recaman sequence for the background music for the OEIS movie was inspired. I've made an alternative musical version of this sequence ( which I would like to believe is also pretty nice. The Recaman sequence is believed to a permutatation of the set of integers The similar sounding Stern Diatomic (my video at: is a permutation of the rationals.

    I don't think that the Catalan numbers are a good canddate for audiblizatiiion. The rise too quickly and anapproach for mapping into an audible

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