Limit A140101(n)/A140100(n) = Tribonacci Constant?

pauldhanna at pauldhanna at
Sun Jun 8 07:52:54 CEST 2008

Consider the complementary sequences A140100 and A140101 
determined by the condition that the term-by-term differences 
and sums must also form a pair of complementary sequences. 
Can anyone find a proof or even a heuristic argument 
to support the conjecture below that the given limits 
involve the tribonacci constant? 
DEFINITION of sequences.  
Least positive integers X(n) and Y(n) > X(n) chosen 
such that X(n) does not appear in 
   {X(k), 1<=k<n}  or  {Y(k), 1<=k<n} 
and Y(n)-X(n) does not appear in 
   {Y(k)-X(k), 1<=k<n}  or  {Y(k)+X(k), 1<=k<n} 
for n>1, starting with X(1)=1 and Y(1)=2. 
The first few terms of the sequences are as follows.  
(x, y)   (y-x, x+y) 
------   ----------
(1, 2)   (1, 3)
(3, 5)   (2, 8)
(4, 8)   (4, 12)
(6, 11)  (5, 17)
(7, 13)  (6, 20)
What are the next terms after this?
Determine the next x to be the least positive integer 
not used earlier as an x or y (here, x = 9), and 
the next y-x value will be the least positive integer 
not appearing earlier as a difference y-x or a sum x+y 
(here, y-x = 7; thus, y = 16 and x+y = 25).
Continuing in this way determines the sequences.  
It would be nice to know the exact behavior of these sequences. 
An interesting concept to consider is the value of certain limits. 
The following limits are given by: 
(1) Limit X(n)/n = 1+1/t 
(2) Limit Y(n)/n = 1+t 
(3) Limit Y(n)/X(n) = t 
(4) Limit [Y(n) + X(n)]/[Y(n) - X(n)] = t^2 
(5) Limit [Y(n)^2 + X(n)^2]/[Y(n)^2 - X(n)^2] = t 
  t = tribonacci constant (A058265) = 1.83928675521416113255...
which solves t^3 = 1 + t + t^2. 
Clearly, given (1) and (2) is true, then (3) is easily deduced, 
while (4) and (5) are true iff t = tribonacci constant. 
The challenge is to show that (1) and (2) are true and further 
that the parameter t is indeed the tribonacci constant. 
If it is easier to prove (3) over (1) and (2), that would be fine; 
I am mainly wanting a justification that t = tribonacci constant. 
The x-values form OEIS entry   A140100: 
The y-values form OEIS entry  A140101: 
See the OEIS entries for 1001 terms computed by Reinhard Zumkeller. 
These terms demonstrate that the tribonacci constant is 
at least near the value of the limit Y(n)/X(n). 
The above sequences can be generated by the following construction. 
Start with an x-y coordinate system and place an 'o' at the origin. 
Define an open position as a point not lying in the same row, column, 
or diagonal (slope +1/-1) as any point previously given an 'o' marker. 
 From then on, place an 'o' marker at the first open position with 
integer coordinates that is nearest the origin and the y-axis in the 
positive quadrant, while simultaneously placing markers at 
rotationally symmetric positions in the remaining three quadrants. 

Example: after the origin, begin placing markers at the coordinates: 
n=1: (1,2), (2,-1), (-1,-2), (-2,1); 
n=2: (3,5), (5,-3), (-3,-5), (-5,3); 
n=3: (4,8), (8,-4), (-4,-8), (-8,4); 
n=4: (6,11),(11,-6),(-6,-11),(-11,6); 
n=5: (7,13),(13,-7),(-7,-13),(-13,7); ...
The result of this process is illustrated in the following diagram. 
Graph: no two points lie in the same row, column, or diagonal. 
Points in the positive quadrant are at (A140100(n), A140101(n)). 
Does the slope of the line nearest the points in the positive quadrant 
tend to the value of the tribonacci constant? 
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