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I get a(8)=18, using the following combinations:<br>
x y x^2 y^2 h^2=x^+y^2 Sorted h^2 Cum# h^2's <br>
<font face="Courier New, Courier">1 0 1 0 1
1 1 <br>
2 0 4 0 4 2 2 <br>
3 0 9 0 9 4 3 <br>
4 0 16 0 16 5 4 <br>
5 0 25 0 25 5 4 <br>
6 0 36 0 36 8 5 <br>
7 0 49 0 49 9 6 <br>
1 1 1 1 2 10 7 <br>
2 1 4 1 5 10 7 <br>
3 1 9 1 10 13 8 <br>
4 1 16 1 17 13 8 <br>
5 1 25 1 26 16 9 <br>
6 1 36 1 37 17 10 <br>
1 2 1 4 5 17 10 <br>
2 2 4 4 8 18 11 <br>
3 2 9 4 13 20 12 <br>
4 2 16 4 20 20 12 <br>
5 2 25 4 29 25 13 <br>
1 3 1 9 10 25 13 <br>
2 3 4 9 13 25 13 <br>
3 3 9 9 18 26 14 <br>
4 3 16 9 25 26 14 <br>
1 4 1 16 17 29 15 <br>
2 4 4 16 20 29 15 <br>
3 4 9 16 25 36 16 <br>
1 5 1 25 26 37 17 <br>
2 5 4 25 29 37 17 <br>
1 6 1 36 37 49 18 <br><br>
</font>Gerald<br><br>
At 03:23 PM 6/6/2005, Hugo Pfoertner wrote:<br>
<blockquote type=cite class=cite cite="">Seqfans, Neil,<br><br>
yesterday I had submitted the new sequence<br><br>
<a href="http://www.research.att.com/projects/OEIS?Anum=A108279" eudora="autourl">http://www.research.att.com/projects/OEIS?Anum=A108279</a><br><br>
(Neil, please keep it, ignoring my request sent by personal
E-mail)<br><br>
%S A108279<br>
1,3,5,8,11,15,18,23,28,33,38,45,51,58,65,73,80,89,97,107,116,126,134,<br>
%T A108279 146,158,169<br>
%N A108279 Number of different sizes occurring among the<br>
A002415(n)=n^2*(n^2-1)/12 squares that can be drawn using points of an
n<br>
X n square array as corners.<br>
%H A108279 H. Bottomley, <a<br>
href="<a href="http://www.research.att.com/~njas/sequences/a2415.gif" eudora="autourl">http://www.research.att.com/~njas/sequences/a2415.gif</a>">Illustration<br>
of initial terms of A002415</a><br>
%e A108279 a(3)=3 because the 6 different squares that can be drawn on
a<br>
3X3 square lattice come in 3 sizes:<br>
%e A108279 4 squares of side length 1:<br>
%e A108279 x.x.o....o.x.x....o.o.o....o.o.o<br>
%e A108279 x.x.o....o.x.x....x.x.o....o.x.x<br>
%e A108279 o.o.o....o.o.o....x.x.o....o.x.x<br>
%e A108279 1 square of side length sqrt(2):<br>
%e A108279 o.x.o<br>
%e A108279 x.o.x<br>
%e A108279 o.x.o<br>
%e A108279 1 square of side length 2:<br>
%e A108279 x.o.x<br>
%e A108279 o.o.o<br>
%e A108279 x.o.x<br>
%e A108279 a(4)=5 because there are 5 different sizes of squares
that<br>
can be drawn using the points of a 4X4 square lattice:<br>
%e A108279 x.x.o.o....o.x.o.o....x.o.x.o....o.x.o.o....x.o.o.x<br>
%e A108279 x.x.o.o....x.o.x.o....o.o.o.o....o.o.o.x....o.o.o.o<br>
%e A108279 o.o.o.o....o.x.o.o....x.o.x.o....x.o.o.o....o.o.o.o<br>
%e A108279 o.o.o.o....o.o.o.o....o.o.o.o....o.o.x.o....x.o.o.x<br>
%Y A108279 Cf. A002415 4-dimensional pyramidal numbers.<br>
%K A108279 more,nonn,new<br>
%O A108279 2,2<br><br>
In a hasty over-reaction to a comment my German friend Rainer
Rosenthal<br>
made in a discussion in the German mathematical newsgroup in a thread
on<br>
square-avoiding lattice colorings I had asked Neil to cancel this<br>
sequence, because there was a deviation from Rainer's results
starting<br>
at a(8)=18 (he suggested a(8)=19). In the meantime I've checked my<br>
program and couldn't find an error.<br><br>
The source code is at<br>
<a href="http://www.randomwalk.de/scimath/nxncol.for" eudora="autourl">http://www.randomwalk.de/scimath/nxncol.for</a><br><br>
Can someone here try to check at least the correctness of
a(8)=18?<br><br>
Thanks<br><br>
Hugo Pfoertner </blockquote></body>
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