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

1,2

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).

Links

Table of n, a(n) for n=1..70.

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

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,... .

Example

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 ...

Crossrefs

Cf. A003269 (row n=1), A134564.

Sequence in context: A353592 A078340 A308541 * A378822 A079297 A276941

Adjacent sequences: A136187 A136188 A136189 * A136191 A136192 A136193

Keyword

nonn,tabl

Author

Clark Kimberling, Dec 20 2007

Status

approved