A342689 Square array read by antidiagonals (upwards): A(n,k) = (k^Fibonacci(n) - 1) / (k - 1) for k >= 0 and n >= 0 with lim_{k -> 1} A(n,k) = A(n,1) = Fibonacci(n).

0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 3, 3, 1, 1, 0, 1, 5, 7, 4, 1, 1, 0, 1, 8, 31, 13, 5, 1, 1, 0, 1, 13, 255, 121, 21, 6, 1, 1, 0, 1, 21, 8191, 3280, 341, 31, 7, 1, 1, 0, 1, 34, 2097151, 797161, 21845, 781, 43, 8, 1, 1, 0, 1, 55, 17179869184, 5230176601, 22369621, 97656, 1555, 57, 9, 1, 1, 0

Offset: 0

Author

Werner Schulte, May 18 2021

Keywords

• nonn
• easy
• tabl

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).

Examples

			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

Cf. A011655 (column k = -1), A057427 (column 0), A000045 (column 1), A063896 (column 2), A000004 (row 0), A000012 (rows 1, 2), A000027 (row 3), A002061 (row 4), A053699 (row 5), A053717 (row 6), A060887 (row 7).

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.