curious eta-GFs and two news seqs

Fri May 16 22:23:31 CEST 2008

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Seqfans,
=20
Let a =3D (1,3,4,6,8,9,11,12,14,16,17,19,21,22,...) =3D A000201
and b =3D (2,5,7,10,13,15,18,20,23,26,28,31,...) =3D A001950.
=20
We can swap selected pairs of terms of a and b so that two nice things
happen:
=20
1.  With each swap both a and b stay monotone (think of them as
"dynamic")
2.  At the end, b consists solely of evens.
=20
Right away, you can look at the above a and b, and swap pairs to get
=20
b =3D (2,4,6,10,12,14,18,20,22,26,...)=20
=20
Can someone generalize?  For example, given m>1 and any k, which Beatty
sequence-pairs allow swapping so that in the end, one of them has all
terms congruent to k mod m?
=20
Clark Kimberling=20
=20

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<DIV><FONT face=3DArial size=3D2><SPAN=20
class=3D049341121-16052008>Seqfans,</SPAN></FONT></DIV>
<DIV><FONT face=3DArial size=3D2><SPAN=20
class=3D049341121-16052008></SPAN></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2><SPAN class=3D049341121-16052008>Let a =3D=
=20
(1,3,4,6,8,9,11,12,14,16,17,19,21,22,...) =3D A000201</SPAN></FONT></DIV>
<DIV><FONT face=3DArial size=3D2><SPAN class=3D049341121-16052008>and b =3D=
=20
(2,5,7,10,13,15,18,20,23,26,28,31,...) =3D A001950.</SPAN></FONT></DIV>
<DIV><FONT face=3DArial size=3D2><SPAN=20
class=3D049341121-16052008></SPAN></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2><SPAN class=3D049341121-16052008>We can sw=
ap selected=20
pairs of terms of a and b so that two nice things happen:</SPAN></FONT></DI=
V>
<DIV><FONT face=3DArial size=3D2><SPAN=20
class=3D049341121-16052008></SPAN></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2><SPAN class=3D049341121-16052008>1.  =
With each=20
swap both a and b stay monotone (think of them as "dynamic")</SPAN></FONT><=
/DIV>
<DIV><FONT face=3DArial size=3D2><SPAN class=3D049341121-16052008>2.  =
At the end,=20
b consists solely of evens.</SPAN></FONT></DIV>
<DIV><FONT face=3DArial size=3D2><SPAN=20
class=3D049341121-16052008></SPAN></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2><SPAN class=3D049341121-16052008>Right awa=
y, you can=20
look at the above a and b, and swap pairs to get</SPAN></FONT></DIV>
<DIV><FONT face=3DArial size=3D2><SPAN=20
class=3D049341121-16052008></SPAN></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2><SPAN class=3D049341121-16052008>b =3D=20
(2,4,6,10,12,14,18,20,22,26,...) </SPAN></FONT></DIV>
<DIV><FONT face=3DArial size=3D2><SPAN=20
class=3D049341121-16052008></SPAN></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2><SPAN class=3D049341121-16052008>Can=20
someone generalize?  For example, given m>1 and any k, which B=
eatty=20
sequence-pairs allow swapping so that in the end, one of them has all terms=
=20
congruent to k mod m?</SPAN></FONT></DIV>
<DIV><FONT face=3DArial size=3D2><SPAN=20
class=3D049341121-16052008></SPAN></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2><SPAN class=3D049341121-16052008>Clark=20
Kimberling </SPAN></FONT></DIV>
<DIV><FONT face=3DArial size=3D2><SPAN=20
class=3D049341121-16052008></SPAN></FONT> </DIV></BODY></HTML>

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