Exploring Wythoff Array

Exploring Wythoff Array


Wythoff array (see also the Wikipedia article for illustration) is an infinite two-dimensional grid of integers that is seeded with 1 and 2 to start off the first row. In each row, each element equals the sum of the previous two elements, so the first row contains precisely the Fibonacci numbers.

The first element of each row after the first is the smallest integer c that does not appear anywhere in the previous rows. Since every row is strictly ascending and grows exponentially fast, you can find this out by looking at relatively short finite prefixes of these rows. To determine the second element of that row, let a and b be the first two elements of the previous row. If the difference c-a equals 2, the second element of that row equals b+3, and otherwise that element equals b+5.

This construction guarantees the Wythoff array to be an interspersion of positive integers; every positive integer will appear exactly once in the entire infinite grid, with no gaps or duplicates anywhere! (This result also nicely highlights the deeper combinatorial importance of the deceptively simple Fibonacci numbers as potential building blocks of integers and their sequences.)

This function should return the position of n in the Wythoff array as tuple (row, col), rows and columns both starting from zero.

Input: Integer (int).

Output: Position as tuple of two integers.


assert wythoff_array(21) == (0, 6)
assert wythoff_array(47) == (1, 5)
assert wythoff_array(1042) == (8, 8)

The mission was taken from Python CCPS 109. It is taught for Ryerson Chang School of Continuing Education by Ilkka Kokkarinen

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