Different squares on an n X n lattice

[Gerald McGarvey]

I get a(8)=18, using the following combinations:
x y x^2 y^2 h^2=x^+y^2 Sorted h^2 Cum# h^2's
1 0  1  0 1   1  1
2 0  4  0 4   2  2
3 0  9  0 9   4  3
4 0 16  0 16  5  4
5 0 25  0 25  5  4
6 0 36  0 36  8  5
7 0 49  0 49  9  6
1 1  1  1  2 10  7
2 1  4  1  5 10  7
3 1  9  1 10 13  8
4 1 16  1 17 13  8
5 1 25  1 26 16  9
6 1 36  1 37 17 10
1 2  1  4  5 17 10
2 2  4  4  8 18 11
3 2  9  4 13 20 12
4 2 16  4 20 20 12
5 2 25  4 29 25 13
1 3  1  9 10 25 13
2 3  4  9 13 25 13
3 3  9  9 18 26 14
4 3 16  9 25 26 14
1 4  1 16 17 29 15
2 4  4 16 20 29 15
3 4  9 16 25 36 16
1 5  1 25 26 37 17
2 5  4 25 29 37 17
1 6  1 36 37 49 18

Gerald

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