A372501 The 2-Zeckendorf array of the second kind, read by upward antidiagonals.

0, 2, 1, 5, 4, 3, 7, 9, 8, 6, 10, 12, 16, 14, 11, 13, 17, 21, 27, 24, 19, 15, 22, 29, 35, 45, 40, 32, 18, 25, 37, 48, 58, 74, 66, 53, 20, 30, 42, 61, 79, 95, 121, 108, 87, 23, 33, 50, 69, 100, 129, 155, 197, 176, 142, 26, 38, 55, 82, 113, 163, 210, 252, 320, 286, 231

Offset: 1

Author

A.H.M. Smeets, May 03 2024

Keywords

• nonn
• tabl

Comments

The 2-Zeckendorf array of the second kind is based on the dual Zeckendorf representation of numbers (see A104326).

Column k contains the numbers whose dual Zeckendorf expansion ends "... 0 1^(k-1)" where ^ denotes repetition.

Rows satisfy this recurrence: T(n,k+1) = T(n,k) + T(n,k-1) + 2 for all n > 0 and k > 1.

As a sequence, the array is a permutation of the nonnegative integers.

As an array, T is an interspersion (hence also a dispersion). This holds as well for all Zeckendorf arrays of the second kind.

In general, for the m-Zeckendorf array of the second kind, the row recursion is given by T(n,k) = T(n,k-1) + T(n,k-m) + m, and the first column represent the "even" numbers.

Examples

			Array begins:
       k=1    2    3    4    5    6    7
      +---------------------------------
  n=1 |  0    1    3    6   11   19   32
  n=2 |  2    4    8   14   24   40   66
  n=3 |  5    9   16   27   45   74  121
  n=4 |  7   12   21   35   58   95  155
  n=5 | 10   17   29   48   79  129  210
  n=6 | 13   22   37   61  100  163  265
  n=7 | 15   25   42   69  113  184  299
The same in dual Zeckendorf form shows the pattern of digit suffixes, for example column k=3 is all numbers ending 011:
          k=1      2       3        4
      +------------------------------
  n=1 |     0      1      11      111
  n=2 |    10    101    1011    10111
  n=3 |   110   1101   11011   110111
  n=4 |  1010  10101  101011  1010111
  n=5 |  1110  11101  111011  1110111
		

Crossrefs

Cf. A104326.

Rows n=1..3: A001911, A019274, A014739.

Columns k=1..3: A090909, A276885, A188012.

Cf. k-th prepended column: A022342 (k=1), A023444 (k=2).

Formula

T(n,1) = A090909(n+1).

T(1,k) = A001911(k-1).

T(2,k) = A019274(k-2).

T(3,k) = A014739(k-1).

T(n,1) = floor((n-1)*phi^2) and T(n,k+1) = floor((T(n,k)+1)*phi) for k > 0, where phi = (1+sqrt(5))/2. This can be considered as an alternative way to define the array.