Limit A140101(n)/A140100(n) = Tribonacci Constant?
pauldhanna at juno.com
pauldhanna at juno.com
Sun Jun 8 07:52:54 CEST 2008
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).
GAPHICAL REPRESENTATION.
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.
----------------+---o------------
--o-------------+----------------
----o-----------+----------------
----------------+--o-------------
--------o-------+----------------
-----------o----+----------------
----------------+o---------------
--------------o-+----------------
++++++++++++++++o++++++++++++++++
----------------+-o--------------
---------------o+----------------
----------------+----o-----------
----------------+-------o--------
-------------o--+----------------
----------------+------------o---
----------------+--------------o-
------------o---+----------------
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?
END.
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