Explain this old one...

[Paul D Hanna]

        From the picture, the sequence
        3 5 6 8 10 12 13 15 17 19 20 22 24... 
appears to be the union of four arithmetic sequences:
  3+7n, 5+7n, 6+7n, 8+7n, for n>=0, and is definitely musical.
But what is the significance of 3,5,6,8?

        Another puzzler here is the number triangle:
 1
 2  3
 4  6  9
 8 16 18  24
16 24 36  54  81
33 48 72 108 162 243
...
which seems to possess mistakes at
    T(3,1)=16?, T(3,3)=24?, T(5,1)=33?, 
that is, if the pattern is supposed to be
 1
 2  3
 4  6  9
 8 12 18  27
16 24 36  54  81
32 48 72 108 162 243
...
but perhaps there is simply a more complex formula?
        Paul

On Tue, 27 May 2003 19:06:15 +1000 Brendan McKay <bdm at cs.anu.edu.au>
writes:
> I was just poking around in the Beinecke Rare Book and Manuscript
> Library Photonegatives Database here:  
>    http://highway49.library.yale.edu/photonegatives/
> and I came across something called Utriusque Cosmi Oppenheim
> dated 1617 which contains the sequence 
>   3 5 6 8 10 12 13 15 17 19 20 22 24.
> Matches in OEIS are
>   A022838 Beatty sequence for sqrt(3)
>   A047329 Congruent to {1, 3, 5, 6} mod 7,
> but I don't know if either of them is the correct sequence.
> 
> A direct link to the manuscript page is
>   
>
http://highway49.library.yale.edu/photonegatives/oneITEM.asp?pid=39002036
242312&iid=3624231
> (click the image for a close view).
> 
> Can anyone solve this?  I guess it is something musical.
> 
> Brendan.
> 
>