A134563 -id:A134563 - cp's OEIS frontend

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

Clark Kimberling, Dec 20 2007

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).

			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 ...
 ...

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

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

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

A134564 Array read by antidiagonals: row n consists of numbers whose 4th-order Zeckendorf representation has exactly n terms.

Original entry on oeis.org

1, 2, 6, 3, 8, 25, 4, 9, 32, 94, 5, 11, 34, 120, 344, 7, 12, 35, 127, 439, 1251, 10, 13, 42, 129, 465, 1596, 4543, 14, 15, 44, 130, 472, 1691

Offset: 1

Clark Kimberling, Nov 01 2007, Dec 18 2007

A permutation of the positive integers.

			Northwest corner:
   1   2   3   4   5   7 10 14 19 26 36 50 69 ...
   6   8   9  11  12  13 ...
  25  32  34  35  42  44 ...
  94 120 127 129 130 156 ...
For example, 32 = 26 + 5 + 1 has 3 terms, so 32 is in row 3.

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

Cf. A003269, A136190, A134563.

Row 1, A035513, is the 4th-order Zeckendorf basis, b(1), b(2), b(3), .... Every positive integer has a unique 4th-order Zeckendorf representation b(i(1)) + b(i(2)) + ... + b(i(n)), where |i(h) - i(j)| >= 4 for distinct h and j.

cp's OEIS Frontend

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

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