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A356969
A(n, k) is the sum of the terms in common in the dual Zeckendorf representations of n and of k; square array A(n, k) read by antidiagonals, n, k >= 0.
2
0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 0, 0, 3, 0, 0, 0, 0, 1, 2, 1, 1, 2, 1, 0, 0, 0, 2, 2, 4, 2, 2, 0, 0, 0, 1, 2, 3, 3, 3, 3, 2, 1, 0, 0, 1, 2, 2, 4, 5, 4, 2, 2, 1, 0, 0, 0, 0, 3, 0, 5, 5, 0, 3, 0, 0, 0, 0, 1, 2, 1, 1, 2, 6, 2, 1, 1, 2, 1, 0
OFFSET
0,13
COMMENTS
The dual Zeckendorf representation corresponds to the lazy Fibonacci representation.
See A334348 for the sequence dealing with Zeckendorf (or greedy Fibonacci) representations. Unlike A334348, the present sequence is not associative.
FORMULA
A(n, k) = A022290(A003754(n+1) AND A003754(k+1)) (where AND denotes the bitwise AND operator, A004198).
A(n, k) = A(k, n).
A(n, 0) = 0.
A(n, n) = n.
EXAMPLE
Square array A(n, k) begins:
n\k| 0 1 2 3 4 5 6 7 8 9 10 11 12 13
---+----------------------------------------------
0| 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1| 0 1 0 1 1 0 1 0 1 1 0 1 1 0
2| 0 0 2 2 0 2 2 2 2 0 2 2 0 2
3| 0 1 2 3 1 2 3 2 3 1 2 3 1 2
4| 0 1 0 1 4 3 4 0 1 4 3 4 4 3
5| 0 0 2 2 3 5 5 2 2 3 5 5 3 5
6| 0 1 2 3 4 5 6 2 3 4 5 6 4 5
7| 0 0 2 2 0 2 2 7 7 5 7 7 0 2
8| 0 1 2 3 1 2 3 7 8 6 7 8 1 2
9| 0 1 0 1 4 3 4 5 6 9 8 9 4 3
10| 0 0 2 2 3 5 5 7 7 8 10 10 3 5
11| 0 1 2 3 4 5 6 7 8 9 10 11 4 5
12| 0 1 0 1 4 3 4 0 1 4 3 4 12 11
13| 0 0 2 2 3 5 5 2 2 3 5 5 11 13
PROG
(PARI) See Links section.
CROSSREFS
KEYWORD
nonn,base,tabl
AUTHOR
Rémy Sigrist, Sep 06 2022
STATUS
approved