A136189 The 3rd-order Zeckendorf array, T(n,k), read by antidiagonals.

1, 2, 5, 3, 8, 7, 4, 12, 11, 10, 6, 17, 16, 15, 14, 9, 25, 23, 22, 21, 18, 13, 37, 34, 32, 31, 27, 20, 19, 54, 50, 47, 45, 40, 30, 24, 28, 79, 73, 69, 66, 58, 44, 36, 26, 41, 116, 107, 101, 97, 85, 64, 53, 39, 29, 60, 170, 157, 148, 142, 125, 94, 77, 57, 43, 33, 88, 249, 230

Offset: 1

Author

Clark Kimberling, Dec 20 2007

Keywords

• nonn
• tabl

Comments

Rows satisfy this recurrence: T(n,k) = T(n,k-1) + T(n,k-3) for all k>=4.

Except for initial terms, (row 1) = A000930 (column 1) = A020942 (column 2) = A064105 (column 3) = A064106.

As a sequence, the array is a permutation of the natural numbers.

As an array, T is an interspersion (hence also a dispersion).

Examples

			Northwest corner:
  1  2  3  4  6  9  13  19 ...
  5  8 12 17 25 37  54  79 ...
  7 11 16 23 34 50  73 107 ...
 10 15 22 32 47 69 101 148 ...
 ...
		

Links

• Clark Kimberling, The Zeckendorf array equals the Wythoff array, Fibonacci Quarterly 33 (1995) 3-8.
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A000930, A035513, A134563, A136175, A136495, A202342.

Formula

Row 1 is the 3rd-order Zeckendorf basis, given by initial terms b(1)=1, b(2)=2, b(3)=3 and recurrence b(k) = b(k-1) + b(k-3) for k>=4. Every positive integer has a unique 3-Zeckendorf representation: n = b(i(1)) + b(i(2)) + ... + b(i(p)), where |i(h)-i(j)| >= 3. Rows of T are defined inductively: T(n,1) is the least positive integer not in an earlier row. T(n,2) is obtained from T(n,1) as follows: if T(n,1) = b(i(1)) + b(i(2)) + ... + b(i(p)), then T(n,k+1) = b(i(1+k)) + b(i(2+k)) + ... + b(i(p+k)) for k=1,2,3,... .

A(n, k) = A000930(k)*A202342(n) + A000930(k-2)*A136495(n) + A000930(k-1)*(n-1) for n > 1. - Alan Michael Gómez Calderón, Dec 23 2024