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

Sun Jun 8 22:13:45 CEST 2008

Seqfans, 
     Here is my intuitive reasoning for the conjecture that 
Limit A140101(n)/A140100(n) = t = tribonacci constant. 

Since A140100 and A140101 are complements of each other,
and since limit A140101(n)/A140100(n) = t  seems to exist, 
one is motivated by the Beatty Theorem to write: 
  A140100(n) = [n*(1+1/t)]
  A140101(n) = [n*(1+t)] ;
(though this is not true for any t, it is very nearly so). 

Likewise, since A140102 and A140103 are complements of each other,
and since limit A140103(n)/A140102(n) = s  seems to exist, 
one is motivated by the Beatty Theorem to write: 
  A140102(n) = [n*(1+1/s)]
  A140103(n) = [n*(1+s)] ;
(though this is not true for any s, it is very nearly so). 

 From the definition of these sequences:
  A140102(n) = A140101(n) - A140100(n) ;
  A140103(n) = A140101(n) + A140100(n). 
So, roughly speaking, 
  [n*(1+1/s)] ~ [n*(1+t)]  - [n*(1+1/t)]  
  [n*(1+s)]    ~ [n*(1+t)]  + [n*(1+1/t)] ;
or,
  1+1/s = (1+t)  - (1+1/t)  
  1+s    = (1+t)  + (1+1/t) 
so that
  1/s = (t^2 - t - 1)/t 
  s    = (t^2 + t + 1)/t .
These equations isolate t to satisfy:
  (t^2 + t + 1)*(t^2 - t - 1) = t^2 
or
  t^4 - 2*t^2 - 2*t - 1  =  (t^3 - t^2 - t - 1)*(1+t)  =  0.

Therefore, I reason that t must be the tribonacci constant, i.e.,
the real-valued t that satisifies:  t^3 = t^2 + t + 1. 

This is not even close to a proof, and abuses the Beatty Theorem,
but it is my intuitive justification for the conjecture. 

An actual proof may be quite involved indeed, but it would be nice.
     Paul 

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. 

On Sun, 8 Jun 2008 01:52:54 -0400 pauldhanna at juno.com writes:
Seqfans, 
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? 

Thanks, 
   Paul 

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. 

EXAMPLE.
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. 

CONJECTURE: 
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 
where 
  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: 
1,3,4,6,7,9,10,12,14,15,17,18,20,21,23,24,26,27,29,30,32,34,
35,37,38,40,41,43,44,46,47,49,51,52,54,55,57,58,60,61,63,64,
66,67,69,71,72,74,75,77,78,80,82,83,85,86,88,89,91,92,94,95,
97,98,100,102,103,105,106,108,109,111,112,114,115,117,119,...

The y-values form OEIS entry  A140101: 
2,5,8,11,13,16,19,22,25,28,31,33,36,39,42,45,48,50,53,56,59,
62,65,68,70,73,76,79,81,84,87,90,93,96,99,101,104,107,110,
113,116,118,121,124,127,130,133,136,138,141,144,147,150,153,
156,158,161,164,167,170,173,175,178,181,184,187,190,193,195,...

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). 
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