OFFSET
1,3
COMMENTS
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.
a(n) is also the number of rectangular isosceles triangles, distinct up to congruence, on an n X n grid (or geoboard). - Martin Renner, May 03 2011
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1000
Henry Bottomley, Illustration of initial terms of A002415
EXAMPLE
a(3)=3 because the 6 different squares that can be drawn on a 3 X 3 square lattice come in 3 sizes:
4 squares of side length 1:
x.x.o o.x.x o.o.o o.o.o
x.x.o o.x.x x.x.o o.x.x
o.o.o o.o.o x.x.o o.x.x
1 square of side length sqrt(2):
o.x.o
x.o.x
o.x.o
1 square of side length 2:
x.o.x
o.o.o
x.o.x
.
a(4)=5 because there are 5 different sizes of squares that can be drawn using the points of a 4 X 4 square lattice:
x.x.o.o o.x.o.o x.o.x.o o.x.o.o x.o.o.x
x.x.o.o x.o.x.o o.o.o.o o.o.o.x o.o.o.o
o.o.o.o o.x.o.o x.o.x.o x.o.o.o o.o.o.o
o.o.o.o o.o.o.o o.o.o.o o.o.x.o x.o.o.x
MATHEMATICA
a[n_] := Module[{v = Table[0, (n - 1)^2]}, Do[v[[k^2 + (w - k)^2]] = 1, {w, 1, n - 1}, {k, 0, w - 1}]; Total[v]]; Array[a, 55](* Jean-François Alcover, Oct 08 2017, after Andrew Howroyd *)
PROG
(PARI)
a(n) = my(v=vector((n-1)^2)); for(w=1, n-1, for(k=0, w-1, v[k^2+(w-k)^2]=1)); vecsum(v); \\ Andrew Howroyd, Sep 17 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Hugo Pfoertner, Jun 05 2005
EXTENSIONS
More terms from David W. Wilson, Jun 07 2005
STATUS
approved