[seqfan] Re: Partition numbers and Fibonacci numbers

[Jonathan Stauduhar]

I think this might help?

http://www.youtube.com/watch?v=5TVOkD7hvCY

On 10/13/2012 2:22 PM, Peter Luschny wrote:
> Today I was looking at this number triangle, but I could
> not make sense of it.
>
>   0:  [ 1]
>   1:  [ 1,   1]
>   2:  [ 2,   2,   2]
>   3:  [ 3,   3,   3,   3]
>   4:  [ 5,   5,   5,   5,   5]
>   5:  [ 7,   8,   8,   8,   8,   8]
>   6:  [11,  14,  13,  13,  13,  13,  13]
>   7:  [15,  23,  22,  21,  21,  21,  21,  21]
>   8:  [22,  39,  36,  35,  34,  34,  34,  34,  34]
>   9:  [30,  65,  60,  57,  56,  55,  55,  55,  55,  55]
> 10:  [42, 109,  99,  94,  91,  90,  89,  89,  89,  89,  89]
> 11:  [56, 183, 164, 154, 149, 146, 145, 144, 144, 144, 144, 144]
>
> In the first column are the partition numbers and on the
> diagonal are the Fibonacci numbers. Can anybody give
> or point to an interpretation of this connection?
>
> The formal definition in `Sage´ is:
>
> @CachedFunction
> def PartToFibo(n, m):
>      if n<  2: return 1
>      S = 0; J = n-1; j = m
>      while 0<  J:
>          T = PartToFibo(J, m)
>          S = S-T if (j//m)%m == 0 else S+T
>          J -= j//m if j%m == 0 else j
>          j += 1
>      return S
>
> for n in (0..12): [PartToFibo(n+1,m+2) for m in (0..n)]
>
> Peter
>
> http://oeis.org/A000041  http://oeis.org/A000045
>
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