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Search: id:A140101
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| A140101 |
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Least positive integer Y(n) > X(n) chosen with the least positive integer X(n) so that neither Y(n) nor X(n) appear earlier in {Y(k),1<=k<n} or {X(k),1<=k<n} 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} for n>1, starting with Y(1)=2 and X(1)=1. |
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+0
5
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| 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
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Compare with A140099(n)=[n*(1+t)], a Beatty sequence involving the
tribonacci constant t = t^3 - t^2 - 1 = 1.83928675521416113255...
Conjecture: A140101(n) - A140099(n) = only 0 or 1 for n>=1;
see A140105 for the positions where a difference of 1 occurs.
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LINKS
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Reinhard Zumkeller, Table of n, Y(n) for n=1..1001
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FORMULA
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Sequence A140100={X(n),n>=1} is the complement of A140101={Y(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 Y(n)/n = 1+t and limit of X(n)/n = 1+1/t
so that 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.
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EXAMPLE
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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.
This sequence gives the y-coordinates of the positive quadrant in the
construction given in the examples for A140100.
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PROGRAM
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(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))))))}
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CROSSREFS
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Cf. A140100 (complement); A140102, A140103, A140105.
Cf. related Beatty sequences: A140098, A140099; A000201.
Cf. A058265 (tribonacci constant).
Sequence in context: A007826 A108589 A093609 this_sequence A141207 A059560 A022842
Adjacent sequences: A140098 A140099 A140100 this_sequence A140102 A140103 A140104
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Jun 04 2008
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EXTENSIONS
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Terms computed independently by Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com) and Joshua Zucker (joshua.zucker(AT)stanfordalumni.org).
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