[seqfan] Gilbreath on Fibonacci

[Sean A. Irvine]

Hi,

In terms of getting some traction on the sequences arising from the
Gilbreath transform, it looks to me like the Gilbreath transform of the
Fibonacci numbers which is apparently 0,1,1 repeated should be easy to
prove. In this case, terms with index >= n in row n, are simply the
Fibonacci numbers themselves and that after each set of three rows we have
one additional copy of 0,1,1 at the start of the row.  So I think you could
make this more formal with some kind of induction looking back 3 rows?

1:  0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
2:  1, 0, 1, 1, 2, 3, 5,  8, 13, 21, 34, 55, ...
3:  1, 1, 0, 1, 1, 2, 3,  5,  8, 13, 21, 34, ...
4:  0, 1, 1, 0, 1, 1, 2,  3,  5,  8, 13, 21, ...
5:  1, 0, 1, 1, 0, 1, 1,  2,  3,  5,  8, 13, ...
6:  1, 1, 0, 1, 1, 0, 1,  1,  2,  3,  5,  8, ...
7:  0, 1, 1, 0, 1, 1, 0,  1,  1,  2,  3,  5, ...
...

Sean.