[seqfan] Re: Partition numbers and Fibonacci numbers

[Aai]

Taking the first differences  of a row (reverse order) shows the 
beginning of a second order fibonacci sequence http://oeis.org/A010049

differences for n=40

       0       0        0        0        1        1 3          5
      10      18       33       59      105      185 324        564
     977    1685     2895     4957     8462    14406     24465 41455
   70104  118337   199438   335674   564424   948661   1595256 2688082
4551158 7778660 13529838 24280308 46013459 95844041 246123529 _610284382

starts to deviate from the 2nd order Fib Seq at 70104 (4th row first 
elem.). It should be 70101.



May be of some help.


On 13-10-12 23:22, 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
>
> _______________________________________________
>
> Seqfan Mailing list -http://list.seqfan.eu/

-- 
Met vriendelijke groet,
@@i = Arie Groeneveld