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A372501
The 2-Zeckendorf array of the second kind, read by upward antidiagonals.
0
0, 2, 1, 5, 4, 3, 7, 9, 8, 6, 10, 12, 16, 14, 11, 13, 17, 21, 27, 24, 19, 15, 22, 29, 35, 45, 40, 32, 18, 25, 37, 48, 58, 74, 66, 53, 20, 30, 42, 61, 79, 95, 121, 108, 87, 23, 33, 50, 69, 100, 129, 155, 197, 176, 142, 26, 38, 55, 82, 113, 163, 210, 252, 320, 286, 231
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
T(n,1) = A090909(n+1).
T(1,k) = A001911(k-1).
T(2,k) = A019274(k-2).
T(3,k) = A014739(k-1).
T(n,1) = floor((n-1)*phi^2) and T(n,k+1) = floor((T(n,k)+1)*phi) for k > 0, where phi = (1+sqrt(5))/2. This can be considered as an alternative way to define the array.
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
Cf. A104326.
Rows n=1..3: A001911, A019274, A014739.
Columns k=1..3: A090909, A276885, A188012.
Cf. k-th prepended column: A022342 (k=1), A023444 (k=2).
Sequence in context: A275131 A280513 A185023 * A061579 A094064 A343809
KEYWORD
nonn,tabl
AUTHOR
A.H.M. Smeets, May 03 2024
STATUS
approved