With the first 5 million elements of Ulam(1,2) I've added a new
significant figure: The Ulam(1,2) density period is P=21.6015845, plus
or minus a few e-7s. It's too far away to be 2765/128, and I don't
see anything very attractive in the ISC.
Here's a plot of u(n) versus u(n-1), mod P. I took out the bucket
elements less than 100, and coded 100-999 and 1000-9999 as 1-9 and
a-j. I've noted the common gaps in parentheses, though I may have
made some errors with that. Note that the density is blank for
residues between 9.0 and -8.0 (or between 0.37 and 0.58 if you prefer
normalized mods).
\ u(n-1) (mod P)
u(n) \-8.0 -4.0 0.0 4.0 8.0
(mod P)\ + + + + +
-8.0 + + + + + + + + + + + +(56,34,12) + + + + + + +
+ (61,39,17) + \\\ + (29,7)+
+ + \\\ + \ 3 + \\+
+ + \21 + \ 6 + 11
+(44,22) + 133 + 119 + 11
+ \\ + 154 + 11b+ +1
+ \1 +(83) 266 + 11b +
+ \2 + \ 287+ 12b +
+ \4 + 1 2b9 13b +
+ 2 \6 + 1 3bb +13b +
-4.0 + + 47+ + \8+ + + + + 3bb + + + + 13b + + + + +
+ 9b 1b +3bb 1 + 113b +
+ bb 2b 2 + 2bb 23+ 113b +
+ bb +2b 16 + 3bb 55 13b +
+ bb 1+ 3b 39 + 2bb 66 13b +
+ 1bb 5 3b 3b+ 3bb +66 39+
(7,27,5) 2bb1b 2b 5b 3bb + 55 36
+ \\\ 3cb1c 1b 5b 3b8+ 65 13
+ \11 4cb1d 1b +7b 296 55 +11
+ \11 5cb1e 7 + 8b 164 55 +
0.0 ++++++++\11++6cb+f+++5+++7b+++++142+++55++++++
+ \ \+16bb g 2 7c + 21 54 +
+ \11 16bb h + 7b + 53+
+ \11 17bb h + 7b + 33
+ \32 17bb h+ 7b + 21
+ 1 +\32 17b8 i 6b+ +11
+ 2 + 142 27b7+i 7b + \\
+ 2 + 31 27b6 h 6b (37,15)
+ 13 + 12 16b5 h +5b +
+ 24+ 1 15b4 h + 59 +
4.0 + + + + + +25 + + + + \594 g+ + + +48 + + + + +
+ 25 +\473 g + 26 +
+ +34 + \473 g + 25 +
+ + 35 + \462 f + 14 +
+ + 24 + \352 e+ 13 +
+ + 23 + \341 e \2+
+ + 11 + \22 d \1
+ + \\ + \1 d \\
+ + (30,8)+ \ d (20,42)
+ + + \ c +
8.0 + + + + + + + + + + + + + + + + \ c+ + + + +
\ b
\ 5
\\\\ \
(69,47,25,3,2)
There definitely seems to be more going on than just a density
attractor near 0 mod P--why are the the dense areas not centered in
the basin? I'm considering a try of this with u(n) versus u(a(n)),
where u(n) = u(a(n)) + u(b(n)) is the canonical decomposition. (I
suppose a(n) < b(n), but I wonder if sometimes the other choice would
be better.)
By the way, I'll warn anyone who wants to work with this period to be
numerically careful. Calculating u(n) mod 21.6015845 in floating
point gets significantly imprecise when u(n) goes above a few million
(Allegro CL on an Ultra). I was about ready to report some curious
granularity effects modulo P/256 but fortunately I checked with
rational arithmetic first. Things got smooth.
Dan
