[seqfan] Some OEIS-related YouTube videos

[Dale Gerdemann]

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