OFFSET
1,2
COMMENTS
The 2-Zeckendorf array of the second kind is based on the dual Zeckendorf representation of numbers (see A104326).
Column k contains the numbers whose dual Zeckendorf expansion ends "... 0 1^(k-1)" where ^ denotes repetition.
Rows satisfy this recurrence: T(n,k+1) = T(n,k) + T(n,k-1) + 2 for all n > 0 and k > 1.
As a sequence, the array is a permutation of the nonnegative integers.
As an array, T is an interspersion (hence also a dispersion). This holds as well for all Zeckendorf arrays of the second kind.
In general, for the m-Zeckendorf array of the second kind, the row recursion is given by T(n,k) = T(n,k-1) + T(n,k-m) + m, and the first column represent the "even" numbers.
FORMULA
EXAMPLE
Array begins:
k=1 2 3 4 5 6 7
+---------------------------------
n=1 | 0 1 3 6 11 19 32
n=2 | 2 4 8 14 24 40 66
n=3 | 5 9 16 27 45 74 121
n=4 | 7 12 21 35 58 95 155
n=5 | 10 17 29 48 79 129 210
n=6 | 13 22 37 61 100 163 265
n=7 | 15 25 42 69 113 184 299
The same in dual Zeckendorf form shows the pattern of digit suffixes, for example column k=3 is all numbers ending 011:
k=1 2 3 4
+------------------------------
n=1 | 0 1 11 111
n=2 | 10 101 1011 10111
n=3 | 110 1101 11011 110111
n=4 | 1010 10101 101011 1010111
n=5 | 1110 11101 111011 1110111
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
A.H.M. Smeets, May 03 2024
STATUS
approved