1,2

Compare with A140098(n)=[n*(1+1/t)], a Beatty sequence involving the

tribonacci constant t = t^3 - t^2 - 1 = 1.83928675521416113255...

Conjecture: A140100(n) - A140098(n) = only 0 or 1 for n>=1;

see A140104 for the positions where a difference of 1 occurs.

Reinhard Zumkeller, Table of n, X(n) for n=1..1001

Sequence A140101={Y(n),n>=1} is the complement of A140100={X(n),n>=1},

while the sequence of differences, A140102={Y(n)-X(n),n>=1}, forms the

complement of the sequence of sums, A140103={Y(n)+X(n),n>=1}.

CONJECTURE: the limit of X(n)/n = 1+1/t and limit of Y(n)/n = 1+t

where the limit of Y(n)/X(n) = t = tribonacci constant (A058265),

and thus the limit of [Y(n) + X(n)]/[Y(n) - X(n)] = t^2

and the limit of [Y(n)^2 + X(n)^2]/[Y(n)^2 - X(n)^2] = t.

Start with X(1)=1, Y(1)=2 ; Y(1)-X(1)=1, Y(1)+X(1)=3.

Next choose X(2)=3 and Y(2)=5 ; Y(2)-X(2)=2, Y(2)+X(2)=8.

Next choose X(3)=4 and Y(3)=8 ; Y(3)-X(3)=4, Y(3)+X(3)=12.

Next choose X(4)=6 and Y(4)=11 ; Y(4)-X(4)=5, Y(4)+X(4)=17.

Continue to choose the least positive X and Y>X not appearing earlier

such that Y-X and Y+X do not appear earlier as a difference or sum.

CONSTRUCTION: PLOT OF (A140100(n), A140101(n)).

This sequence gives the x-coordinates of 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 x-y 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)).

A140101 begins: [2,5,8,11,13,16,19,22,25,28,31,33,36,39,42,...].

(PARI) /* Print (x, y) coordinates of the positive quadrant */

Cf. A140101 (complement); A140102, A140103, A140104.

Cf. related Beatty sequences: A140098, A140099; A000201.

Cf. A058265 (tribonacci constant).

Sequence in context: A103877 A072561 A141206 this_sequence A093610 A140758 A094178

Adjacent sequences: A140097 A140098 A140099 this_sequence A140101 A140102 A140103

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Jun 04 2008

Terms computed independently by Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com) and Joshua Zucker (joshua.zucker(AT)stanfordalumni.org).