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A136189
The 3rd-order Zeckendorf array, T(n,k), read by antidiagonals.
15
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
1,2
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).
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,... .
EXAMPLE
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 ...
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Dec 20 2007
STATUS
approved