OFFSET
0,12
COMMENTS
Replacing Fibonacci(n), A000045, with Lucas(n), A000032, you get another square array B(n,k). The terms satisfy the same recurrence equation B(n,k) = (k-1) * (B(n-1,k) * B(n-2,k) + B(n-1,k) + B(n-2,k) for k >= 0 and n > 1 with initial values B(0,k) = k+1 and B(1,k) = 1. Please take account of lim_{k -> 1} (k^Lucas(n) - 1) / (k - 1) = Lucas(n).
FORMULA
A(n,k) = (k - 1) * A(n-1,k) * A(n-2,k) + A(n-1,k) + A(n-2,k) for k >= 0 and n > 1 with initial values A(0,k) = 0 and A(1,k) = 1.
EXAMPLE
The array A(n,k) for k >= 0 and n >= 0 begins:
n \ k: 0 1 2 3 4 5 6 7 8 9 10 11
=========================================================================
0 : 0 0 0 0 0 0 0 0 0 0 0 0
1 : 1 1 1 1 1 1 1 1 1 1 1 1
2 : 1 1 1 1 1 1 1 1 1 1 1 1
3 : 1 2 3 4 5 6 7 8 9 10 11 12
4 : 1 3 7 13 21 31 43 57 73 91 111 133
5 : 1 5 31 121 341 781 1555 2801
6 : 1 8 255 3280 21845 97656
7 : 1 13 8191 797161 22369621
8 : 1 21 2097151 5230176601
9 : 1 34 17179869184
10 : 1 55
11 : 1 89
etc.
CROSSREFS
KEYWORD
AUTHOR
Werner Schulte, May 18 2021
STATUS
approved