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Wythoff Array

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The Wythoff array is an interspersion array that can be constructed by beginning with the Fibonacci numbers {F_2,F_3,F_4,F_5,...} in the first row and then building up subsequent rows by iteratively adding {F_(3+k),F_(4+k),F_(5+k),F_(6+k),...}, where k=0 or 1 is the smallest offset producing an initial term that has not occurred in a previous row. This process gives the array

1 2 3 5 8 13 21 34 55 ...; 4 7 11 18 29 47 76 123 199 ...; 6 10 16 26 42 68 110 178 288 ...; 9 15 24 39 63 102 165 267 432 ...; 12 20 32 52 84 136 220 356 576 ...; 14 23 37 60 97 157 254 411 665 ...; 17 28 45 73 118 191 309 500 809 ...; 19 31 50 81 131 212 343 555 898 ...; 22 36 58 94 152 246 398 644 1042 ...; | | | | | | | | | ....
(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 nth row given by

a_(n1)=|_nphi_|+n-1
(2)
=|_|_nphi_|phi_|,
(3)

where phi is the golden ratio and |_x_| is the floor function. The offset k is used to pass from a row to the next row. For n>=1, rows numbered |_nphi^2_|, i.e., 2, 5, 7, 10, 13, ... (OEIS A001950) have outgoing offset k=0, while rows numbered |_nphi_|, i.e., 1, 3, 4, 6, 8, ... (OEIS A000201) have outgoing offset k=1.

For n,k>=1, the element a_(nk) can be given explicitly by

a_(nk)=(n-1)F_k+F_(k+1)|_nphi_|,
(4)

where F_k is the kth Fibonacci number, with F_0=0 and F_1=1.

See also

Beatty Sequence, Fibonacci Number, Interspersion, Stolarsky Array

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References

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."

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Wythoff Array

Cite this as:

Weisstein, Eric W. "Wythoff Array." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/WythoffArray.html

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