[seqfan] Re: Lattice Squares

[Richard Mathar]

Continuing on 
http://list.seqfan.eu/pipermail/seqfan/2009-July/001915.html

ftaw> From seqfan-bounces at list.seqfan.eu Sat Jul 18 18:36:00 2009
ftaw> To: seqfan at list.seqfan.eu
ftaw> Date: Sat, 18 Jul 2009 12:28:35 -0400
ftaw> From: franktaw at netscape.net
ftaw> Subject: [seqfan]  Lattice Squares
ftaw> ...
ftaw> Richard, you appear to be assuming that the squares are aligned with 
ftaw> the lattice.  If we don't make this assumption, there are two more for 
ftaw> n=2:
ftaw> 
ftaw> .X...
ftaw> ....X
ftaw> .....
ftaw> X....
ftaw> ...X.
ftaw> 
ftaw> and its mirror image.

...this explains all of the difference. If I run a blind full
search including squares with non-horizontal or non-vertical sides,
I also get 1,8,33,96,225,456,...,53600

myplot := proc(xNE,yNE,xNW,yNW,xSW,ySW,xSE,ySE,n)
	for y from  n to -n by -1 do
	for x from  -n to n do
		if x=xNE and y = yNE then
			printf("x") ;
		elif x=xNW and y = yNW then
			printf("x") ;
		elif x=xSW and y = ySW then
			printf("x") ;
		elif x=xSE and y = ySE then
			printf("x") ;
		else
			printf(".") ;
		fi;

	od:
	printf("\n") ;
	od:
	printf("\n\n") ;
end:

sqs := proc(n)
	a := 0 ;
	for xNE from 1 to n do
		for yNE from 1 to n do
			for xNW from -n to -1 do
			for yNW from 1 to n do
				xSW := xNW-(yNW-yNE) ;
				ySW := yNW-(xNE-xNW) ;
				if xSW < 0 and xSW >= -n and ySW < 0 and ySW >= -n then
					xSE := -xNW+xSW+xNE ;
					ySE := -yNW+ySW+yNE ;
					if xSE > 0 and xSE <= n and ySE < 0 and ySE >= -n then
					# myplot(xNE,yNE,xNW,yNW,xSW,ySW,xSE,ySE,n) ;
					a := a+1 ;
					fi;
				fi;
			od:
			od:
		od:
	od:
	a ;
end:
for n from 1 to 20 do
	print(n, sqs(n)) ;
od:

Richard J. Mathar