[seqfan] Re: Partition numbers and Fibonacci numbers

[Peter Luschny]

JS> I think this might help?
JS> http://www.youtube.com/watch?v=5TVOkD7hvCY

Thanks, that's a nice video. I know the interpretation of T[n,0]
(partitions of n) and of T[n,n] (compositions of n+2 not using 1's),
but what is, for example T[10,3]?

SW> A table (in the attachment) with the values of S, J, j, T...
SW> during the iteration for m = 2 (this is m = 0 in the number
SW> triangle) also seems to show a connection to the Fibonacci
SW> sequence.

Yes, it certainly does. m=2 is the base case, Euler's pentagonal
number theorem. In http://oeis.org/A001318 look up the comment of
R. K. Guy which exactly describes the algorithm which I use in a
generalized form. The algorithm just introduces a parameter into
Euler's pentagonal number theorem in such a way that it generates,
left to right in the rows, the number of the partitions of n [....] the
number of compositions not using 1's. Now what is the meaning of
the values in between? I think that I am missing something obvious.

PL>   0:  [ 1]
PL>   1:  [ 1,   1]
PL>   2:  [ 2,   2,   2]
PL>   3:  [ 3,   3,   3,   3]
PL>   4:  [ 5,   5,   5,   5,   5]
PL>   5:  [ 7,   8,   8,   8,   8,   8]
PL>   6:  [11,  14,  13,  13,  13,  13,  13]
PL>   7:  [15,  23,  22,  21,  21,  21,  21,  21]
PL>   8:  [22,  39,  36,  35,  34,  34,  34,  34,  34]
PL>   9:  [30,  65,  60,  57,  56,  55,  55,  55,  55,  55]
PL>  10:  [42, 109,  99,  94,  91,  90,  89,  89,  89,  89,  89]

PL> In the first column are the partition numbers and on the
PL> diagonal are the Fibonacci numbers. Can anybody give
PL> or point to an interpretation of this connection?