# [seqfan] Records in a Product Involving Wythoff Numbers

Paul D Hanna pauldhanna at juno.com
Tue Dec 28 07:22:25 CET 2010

```SeqFans,

Define the characteristic functions WL(x) and WU(x) for the Lower (A000201)
and Upper (A001950) Wythoff sequences, respectively, by:

* WL(x) = 1 + x + x^3 + x^4 + x^6 + x^8 + x^9 + x^11 + x^12 +...+ x^[n*phi] +...

* WU(x) = 1 + x^2 + x^5 + x^7 + x^10 + x^13 + x^15 + x^18 +...+ x^[n*(phi+1)] +...

where phi = (1+sqrt(5))/2, and we included the constant term x^0.

Consider the product WL(-x)*WU(x), in which the coefficients begin:
[1,-1,1,-2,1,0,1,1,0,0,1,-1,1,1,1,2,-1,1,1,0,1,-1,1,1,0,0,1,
-1,1,-2,1,0,1,-1,1,-2,1,-3,1,-1,1,0,1,-1,1,-2,1,0,1,1,0,0,1,
-1,1,-2,1,0,1,-1,1,-2,1,-3,1,-1,2,-2,1,-3,1,-4,1,-2,1,-1,2, ...]

Positions of records for positive coefficients in WL(-x)*WU(x) begin:
1: 0
2: 15
3: 159
4: 303
5: 2887
6: 5471
7: 51839
8: 98207
...

Positions of records for negative coefficients in WL(-x)*WU(x) begin:
-1: 1
-2: 3
-3: 37
-4: 71
-5: 681
-6: 1291
-7: 12237
-8: 23183
-9: 219601
...

Now compare the above positions to A059973:

1,1, 2,4, 9,17, 38,72, 161,305, 682,1292, 2889,5473, 12238,23184, 51841,98209, 219602,416020, 930249,1762289, ...

and we see a coincidence that would lead us to think we have the formula:
* [x^(A059973(4n+1)-2)] WL(-x)*WU(x) = 2n-1 for n>=1;
* [x^(A059973(4n+2)-2)] WL(-x)*WU(x) = 2n for n>=1;
* [x^(A059973(4n-1)-1)] WL(-x)*WU(x) = -(2n-1) for n>=1;
* [x^(A059973(4n)-1)] WL(-x)*WU(x) = -(2n) for n>=1;
(there may be a more concise expression).

If the pattern continues, then we predict that
* the first  '+9' coefficient will occur at position 930247
* the first '-10' coefficient will occur at position 416019
in the product WL(-x)*WU(x).

Could someone verify these numbers?

Thanks,
Paul

```