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<DIV>Seqfans, <BR>Consider the complementary sequences A140100
and A140101 <BR>determined by the condition that the term-by-term
differences </DIV>
<DIV>and sums must also form a pair of complementary sequences.
<BR> <BR>Can anyone find a proof or even a heuristic argument </DIV>
<DIV>to support the conjecture below that the given limits <BR>involve the
tribonacci constant? <BR> <BR>Thanks, <BR> Paul
<BR> <BR>DEFINITION of sequences. <BR>Least positive integers X(n)
and Y(n) > X(n) chosen <BR>such that X(n) does not appear in <BR>
{X(k), 1<=k<n} or {Y(k), 1<=k<n} <BR>and Y(n)-X(n) does
not appear in <BR> {Y(k)-X(k), 1<=k<n} or
{Y(k)+X(k), 1<=k<n} <BR>for n>1, starting with X(1)=1 and Y(1)=2.
<BR> <BR>EXAMPLE.<BR>The first few terms of the sequences are as
follows. <BR>(x, y) (y-x, x+y) <BR>------
----------<BR>(1, 2) (1, 3)<BR>(3, 5) (2, 8)<BR>(4,
8) (4, 12)<BR>(6, 11) (5, 17)<BR>(7, 13) (6, 20)<BR>What
are the next terms after this?<BR>Determine the next x to be the least positive
integer <BR>not used earlier as an x or y (here, x = 9), and <BR>the next y-x
value will be the least positive integer <BR>not appearing earlier as a
difference y-x or a sum x+y <BR>(here, y-x = 7; thus, y = 16 and x+y =
25).<BR>Continuing in this way determines the sequences.
<BR> <BR>It would be nice to know the exact behavior of these
sequences. <BR>An interesting concept to consider is the
value of certain limits. <BR> <BR>CONJECTURE: <BR>The following
limits are given by: <BR>(1) Limit X(n)/n = 1+1/t <BR>(2) Limit Y(n)/n =
1+t <BR>(3) Limit Y(n)/X(n) = t <BR>(4) Limit [Y(n) + X(n)]/[Y(n) - X(n)] = t^2
<BR>(5) Limit [Y(n)^2 + X(n)^2]/[Y(n)^2 - X(n)^2] = t <BR>where <BR> t =
tribonacci constant (A058265) = 1.83928675521416113255...<BR>which solves t^3 =
1 + t + t^2. <BR> <BR>Clearly, given (1) and (2) is true, then (3) is
easily deduced, <BR>while (4) and (5) are true iff t = tribonacci constant.
<BR> <BR>The challenge is to show that (1) and (2) are true and further
<BR>that the parameter t is indeed the tribonacci constant. </DIV>
<DIV> </DIV>
<DIV>If it is easier to prove (3) over (1) and (2), that would be
fine; </DIV>
<DIV>I am mainly wanting a justification that t = tribonacci constant.
<BR> <BR>The x-values form OEIS entry A140100:
<BR>1,3,4,6,7,9,10,12,14,15,17,18,20,21,23,24,26,27,29,30,32,34,<BR>35,37,38,40,41,43,44,46,47,49,51,52,54,55,57,58,60,61,63,64,<BR>66,67,69,71,72,74,75,77,78,80,82,83,85,86,88,89,91,92,94,95,<BR>97,98,100,102,103,105,106,108,109,111,112,114,115,117,119,...<BR> <BR>The
y-values form OEIS entry A140101:
<BR>2,5,8,11,13,16,19,22,25,28,31,33,36,39,42,45,48,50,53,56,59,<BR>62,65,68,70,73,76,79,81,84,87,90,93,96,99,101,104,107,110,<BR>113,116,118,121,124,127,130,133,136,138,141,144,147,150,153,<BR>156,158,161,164,167,170,173,175,178,181,184,187,190,193,195,...<BR> </DIV>
<DIV>See the OEIS entries for 1001 terms computed by <FONT
face="Courier New">Reinhard Zumkeller. </FONT></DIV>
<DIV><FONT face="Courier New">These terms demonstrate that the tribonacci
constant </FONT><FONT face="Courier New">is </FONT></DIV>
<DIV><FONT face="Courier New">at least near the value of
the limit Y(n)/X(n). </FONT></DIV>
<DIV> </DIV>
<DIV> <BR>GAPHICAL REPRESENTATION. <BR>The above sequences can be generated
by the following construction. <BR> <BR>Start with an x-y coordinate system
and place an 'o' at the origin. <BR>Define an open position as a point not lying
in the same row, column, <BR>or diagonal (slope +1/-1) as any point previously
given an 'o' marker. <BR>From then on, place an 'o' marker at the first open
position with <BR>integer coordinates that is nearest the origin and the y-axis
in the <BR>positive quadrant, while simultaneously placing markers at
<BR>rotationally symmetric positions in the remaining three quadrants. </DIV>
<DIV> </DIV>
<DIV>Example: after the origin, begin placing markers at the coordinates:
<BR>n=1: (1,2), (2,-1), (-1,-2), (-2,1); <BR>n=2: (3,5), (5,-3), (-3,-5),
(-5,3); <BR>n=3: (4,8), (8,-4), (-4,-8), (-8,4); <BR>n=4:
(6,11),(11,-6),(-6,-11),(-11,6); <BR>n=5: (7,13),(13,-7),(-7,-13),(-13,7);
...<BR>The result of this process is illustrated in the following diagram.
<BR>----------------+---o------------<BR>--o-------------+----------------<BR>----o-----------+----------------<BR>----------------+--o-------------<BR>--------o-------+----------------<BR>-----------o----+----------------<BR>----------------+o---------------<BR>--------------o-+----------------<BR>++++++++++++++++o++++++++++++++++
<BR>----------------+-o--------------<BR>---------------o+----------------<BR>----------------+----o-----------<BR>----------------+-------o--------<BR>-------------o--+----------------<BR>----------------+------------o---<BR>----------------+--------------o-<BR>------------o---+----------------<BR>Graph:
no two points lie in the same row, column, or diagonal. <BR>Points in the
positive quadrant are at (A140100(n), A140101(n)). </DIV>
<DIV> </DIV>
<DIV>Does the slope of the line nearest the points in the
positive quadrant </DIV>
<DIV>tend to the value of the tribonacci constant? <BR> <BR>END. </DIV>
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