A136190 The 4th-order Zeckendorf array, T(n,k), read by antidiagonals.

1, 2, 6, 3, 9, 8, 4, 13, 12, 11, 5, 18, 17, 16, 15, 7, 24, 23, 22, 21, 20, 10, 33, 31, 30, 29, 28, 25, 14, 46, 43, 41, 40, 39, 35, 27, 19, 64, 60, 57, 55, 54, 49, 38, 32, 26, 88, 83, 79, 76, 74, 68, 53, 45, 34, 36, 121, 114, 109, 105, 102, 93, 73, 63, 48, 37, 50, 167, 157, 150

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-4) for all k>=5.

Except for initial terms, (row 1) = A003269 (row 2) = A014101.

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  5  7 10  14 ...
   6  9 13 18 24 33 46  64 ...
   8 12 17 23 31 43 60  83 ...
  11 16 22 30 41 57 79 109 ...
		

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. A003269 (row n=1), A134564.

Formula

Row 1 is the 4th-order Zeckendorf basis, given by initial terms b(1)=1, b(2)=2, b(3)=3, b(4)=4 and recurrence b(k) = b(k-1) + b(k-4) for k>=5. Every positive integer has a unique 4-Zeckendorf representation: n = b(i(1)) + b(i(2)) + ... + b(i(p)), where |i(h) - i(j)| >= 4. 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,... .