A Wythoff sequence is one of the two complementary Beatty sequences
 |
where 
,
is the golden
ratio, and 
is the floor function. The lower Wythoff sequence
begins 1, 3, 4, 6, 8, 9, 11, 12, 14, 16, ... (OEIS A000201),
and the upper Wythoff sequence begins 2, 5, 7, 10, 13, 15, 18, 20, 23, 26, ... (OEIS
A001950). The pairs 
are the nonzero safe positions
of Wythoff's game, with the smaller heap first.
For a positive integer 
, let 
count representations 
with 
. Ashburn (2026) proved that 
iff 
is twice a Fibonacci number
of value at least two, and 
iff 
is a Lucas number of value
at least seven. The previously known condition 
is that 
is a Fibonacci number
of value at least two. Ashburn (2026) supplies Walnut verification files and reports
Claude assistance with literature searches, computations, and drafting, with the
proofs and computations checked by the author.
