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A140100 Start with Y(0)=0, X(1)=1, Y(1)=2. For n > 1, choose least positive integers Y(n) > X(n) such that neither Y(n) nor X(n) appear in {Y(k), 1 <= k < n} or {X(k), 1 <= k < n} and such that Y(n) - X(n) does not appear in {Y(k) - X(k), 1 <= k < n} or {Y(k) + X(k), 1 <= k < n}; sequence gives X(n) (for Y(n) see A140101). 25
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Sequence A140101 = {Y(n), n >= 1} is the complement of the current sequence, 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}.

Compare with A140098(n) = floor(n*(1+1/t)), a Beatty sequence involving the tribonacci constant t = t^3 - t^2 - 1 = 1.83928675521416113255...

Conjecture: A140100(n) - A140098(n) = A276404(n) is always 0 or 1; see A276406 for the positions where a difference of 1 occurs.

This is the same problem as the "Greedy Queens in a spiral" problem described in A273059. See the Dekking et al. paper and comments in A140101. - N. J. A. Sloane, Aug 30 2016

LINKS

N. J. A. Sloane, Table of n, a(n) for n = 1..50000, Sep 13 2016 (First 1001 terms from Reinhard Zumkeller)

F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), #P1.52.

Eric Duchêne and Michel Rigo, A morphic approach to combinatorial games: the Tribonacci case. RAIRO - Theoretical Informatics and Applications, 42, 2008, pp 375-393. doi:10.1051/ita:2007039. [Also available here]

Robbert Fokkink, Dan Rust, A modification of Wythoff's Nim, arXiv:1904.08339 [math.CO], 2019.

N. J. A. Sloane, Maple program for A140100, A140101, A140102, A140103

FORMULA

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.

From Michel Dekking, Mar 16 2019: (Start)

It is conjectured in A305392 that the first differences of (X(n)) as a word are given by  212121 delta(x), where  x is the tribonacci word x = A092782, and delta is the morphism

      1 -> 2212121212121,

      2 -> 22121212121,

      3 -> 2212121.

This conjecture implies the frequency conjectures above: let N(i,n) be the number of letters i in x(1)x(2)...x(n). Then simple counting gives

       X(13*N(1,n)+11*N(2,n)+7*N(3,n)) = 20*N(1,n)+17*N(2,n)+11*N(3,n), where we neglected the first 6 symbols of X.

It is well known (see, e.g., A092782) that the frequencies of 1, 2 and 3 in x are respectively 1/t, 1/t^2 and 1/t^3. Dividing all the N(i,n) by n, and letting n tend to infinity, we then have to see that

     20*1/t + 17*1/t^2 + 11*1/t^3 = (1+1/t)*(13*1/t + 11*1/t^2 + 7*1/t^3)

This is a simple verification.

End)

EXAMPLE

Start with Y(0)=0, 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 (see A273059 for an equivalent picture - N. J. A. Sloane, Aug 30 2016).

----------------+---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, diagonal, or antidiagonal.

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,...].

MAPLE

See link.

MATHEMATICA

y[0] = 0; x[1] = 1; y[1] = 2;

x[n_] := x[n] = For[yn = y[n - 1] + 1, True, yn++, For[xn = x[n - 1] + 1, xn < yn, xn++, xx = Array[x, n - 1]; yy = Array[y, n - 1]; If[FreeQ[xx, xn] && FreeQ[xx, yn] && FreeQ[yy, xn] && FreeQ[yy, yn] && FreeQ[yy - xx, yn - xn] && FreeQ[yy + xx, yn - xn], y[n] = yn; Return[xn]]]];

Table[x[n], {n, 1, 100}] (* Jean-François Alcover, Jun 17 2018 *)

PROG

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

{X=[1]; Y=[2]; D=[1]; S=[3]; print1("["X[1]", "Y[1]"], "); for(n=1, 100, for(j=2, 2*n, if(setsearch(Set(concat(X, Y)), j)==0, Xt=concat(X, j); for(k=j+1, 3*n, if(setsearch(Set(concat(Xt, Y)), k)==0, if(setsearch(Set(concat(D, S)), k-j)==0, if(setsearch(Set(concat(D, S)), k+j)==0, X=Xt; Y=concat(Y, k); D=concat(D, k-j); S=concat(S, k+j); print1("["X[ #X]", "Y[ #Y]"], "); break); break))))))}

CROSSREFS

Cf. A140101 (complement); A140102, A140103, A276404, A276405, A276406, A275926.

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

Cf. A058265 (tribonacci constant).

Cf. Greedy Queens in a spiral, A273059.

For first difference of A140100, A140101, A140102, A140103 see A305392, A305374, A305393, A305394.

Sequence in context: A072561 A330176 A141206 * A292642 A249117 A093610

Adjacent sequences:  A140097 A140098 A140099 * A140101 A140102 A140103

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 04 2008

EXTENSIONS

Terms computed independently by Reinhard Zumkeller and Joshua Zucker

Edited by N. J. A. Sloane, Aug 30 2016

STATUS

approved

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Last modified April 9 20:55 EDT 2020. Contains 333363 sequences. (Running on oeis4.)