[seqfan] Re: Records in a Product Involving Wythoff Numbers

[Douglas McNeil]

I can't say whether they're _records_, but I confirm (subject to the
usual disclaimers) that your formulae

> * [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;

hold for at least the next few cases:
# [locations, observed, predicted]
(7, 15, 1, 3) [1, 2, -1, -2] [1, 2, -1, -2]
(159, 303, 37, 71) [3, 4, -3, -4] [3, 4, -3, -4]
(2887, 5471, 681, 1291) [5, 6, -5, -6] [5, 6, -5, -6]
(51839, 98207, 12237, 23183) [7, 8, -7, -8] [7, 8, -7, -8]
(930247, 1762287, 219601, 416019) [9, 10, -9, -10] [9, 10, -9, -10]
(16692639, 31622991, 3940597, 7465175) [11, 12, -11, -12] [11, 12, -11, -12]

but so far I'm finding

(299537287, 567451583, 70711161, 133957147) [9, 9, -13, -14] [13, 14, -13, -14]

Possibilities include: (1) Even though it looks like I should have
several digits of accuracy to spare, I've hit a precision issue
between 133957147 and 299537287.  I checked my 299537287 value at 512
bits, which should be more than sufficient, but I might have missed
something.  (2) There's a bug in my code which only manifests at
larger n, all too possible; I tried it a few ways but they weren't
independent of each other as I could only think of one reasonably
efficient approach.  (3) The agreement actually does break down.


Doug

-- 
Department of Earth Sciences
University of Hong Kong