[seqfan] Some OEIS-related YouTube videos
Dale Gerdemann
dale.gerdemann at googlemail.com
Sun Jul 24 21:44:00 CEST 2011
Hello all,
I've just put a sequence of videos on YouTube which are related to
OEIS sequences. Here's the short description that I wrote for YouTube:
The Gold-Silver-Copper-Bronze-Plastic sequence of videos is based
on generalizations of the Golden Ratio Division algorithm shown in
http://www.youtube.com/watch?v=eOxuQQlYG94
to other sequences:
Gold: Fibonacci A000045 http://bit.ly/mRpdk1
Silver: Pell A000129 http://bit.ly/nxDMMf
Copper: A002605 http://bit.ly/nuigqH
Bronze: A001906 http://bit.ly/mY0qIg
Plastic: Padovan A000931 http://bit.ly/o8ZKdK
I use the two numbers after each subtraction both for the x,y
coordinates and for the notes for two musical instruments. In the
case of Plastic, however, there are three numbers after each
subtraction. So I used the extra variable to determine the color,
and a third instrument of course.
Here are some examples of the generalization. Some of the steps are
not immediately apparent, though all can be proved combinatorially as
in D. Gerdemann. Combinatorial proofs of Zeckendorf family
identities. Fibonacci Quarterly, pages 249--262, August, 2008/2009.
Silver example:
3 3 4 2 3
----------------
11)22 11
12 5
-----
10 6
-------->
26 10
12 5
-----
14 5
-------->
33 14
29 12
-----
4 2
-------->
10 4
5 2
-----
5 2
------->
12 5
12 5
----
0 0
Copper example ("copper" and "bronze" are non-standard names)
2 2 1 2
---------
4)8 8
6 4
---
2 4
----->
8 4
6 4
---
2 0
----->
4 4
2 2
---
2 2
----->
6 4
6 4
---
0 0
bronze example
3 2 1
-----------
8)24 -8
21 -8
-----
3 0
-------->
9 -3
8 -3
-----
1 0
-------->
3 -1
3 -1
-----
0 0
Plastic example (the 0 above the 5 is explained below)
0
_5_x__x__x__x__x__0__x__x__x__x__x__0
4)0 4 4
2 2 1
-----
-2 2 3
1 0 0
-----
-3 2 3
----------->
2 0 -3
----------->
0 -1 2
---------->
-1 2 0
----------->
2 -1 -1
----------->
-1 1 2
------------->
1 1 -1
1 0 0
--------
0 1 -1
---------->
1 -1 0
------------>
-1 1 1
------------>
1 0 -1
----------->
0 0 1
-------->
0 1 0
------------>
1 0 0
1 0 0
--------
0 0 0
The 0 above the 5 is the result of two trial divisions with no
intervening shift. It acts simply as another element in the first
column.
So we see that 4a(n) = a(n+4) + a(n-1) + a(n-7) + a(n-13) is a valid
identity for the Padovan numbers.
I hope these are of interest to seqfans. Comments and criticism are
welcome. There are also a couple of alternative versions on my
open-to-the-world Facebook page: http://www.facebook.com/dale.gerdemann.
Dale Gerdemann
Dept. Linguistics
Section for Computational Linguistics
University of Tübingen
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