[seqfan] Partition numbers and Fibonacci numbers
Peter Luschny
peter.luschny at gmail.com
Sat Oct 13 23:22:32 CEST 2012
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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