The Wythoff array is an interspersionarray that can be constructed by beginning with the Fibonacci
numbers
in the first row and then building up subsequent rows by iteratively adding ,
where
or 1 is the smallest offset producing an initial term that has not occurred in a
previous row. This process gives the array
(1)
Read by skew diagonals from lower left to upper right, this gives the sequence 1; 4, 2; 6, 7, 3; ...
(OEIS A083412), while read by skew
diagonals from upper right to lower left, this gives 1; 2, 4; 3, 7, 6; ... (OEIS
A035513).
The first column is given by 1, 4, 6, 9, 12, 14, 17, ... (OEIS A003622), with the initial term of the th row given by
(2)
(3)
where
is the golden ratio and is the floor function.
The offset
is used to pass from a row to the next row. For , rows numbered , i.e., 2, 5, 7, 10, 13, ... (OEIS A001950)
have outgoing offset ,
while rows numbered ,
i.e., 1, 3, 4, 6, 8, ... (OEIS A000201) have
outgoing offset .
Fraenkel, A. and Kimberling, C. "Generalized Wythoff Arrays, Shuffles and Interspersions." Disc. Math.126, 137-149,
1994.Kimberling, C. "Stolarsky Interspersions." Ars Combin.39,
129-138, 1995.Kimberling, C. "Fractal Sequences and Interspersions."
Ars Combin.45, 157-168, 1997.Kimberling, C. "Interspersions
and Dispersions." https://faculty.evansville.edu/ck6/integer/intersp.html.Sloane,
N. J. A. "My Favorite Integer Sequences." In Sequences and
Their Applications (Proceedings of SETA '98) (Ed. C. Ding, T. Helleseth,
and H. Niederreiter). London, England: Springer-Verlag, pp. 103-130, 1999.
https://neilsloane.com/doc/sg.pdf.Sloane,
N. J. A. "The Wythoff Array and the Para-Fibonacci Sequence."
https://oeis.org/classic.html.Sloane,
N. J. A. Sequences A000201/M2322,
A001950/M1332, A003622/M3278,
A035513, and A083412
in "The On-Line Encyclopedia of Integer Sequences."
in the first row and then building up subsequent rows by iteratively adding 
,
where 
or 1 is the smallest offset producing an initial term that has not occurred in a
previous row. This process gives the array
th row given by
is the golden ratio and 
is the floor function.
The offset 
is used to pass from a row to the next row. For 
, rows numbered 
, i.e., 2, 5, 7, 10, 13, ... (OEIS A001950)
have outgoing offset 
,
while rows numbered 
,
i.e., 1, 3, 4, 6, 8, ... (OEIS A000201) have
outgoing offset 
.
,
the element 
can be given explicitly by
is the 
th
Fibonacci number, with 
and 
.
