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A108279
a(n) = number of squares with corners on an n X n grid, distinct up to congruence.
8
0, 1, 3, 5, 8, 11, 15, 18, 23, 28, 33, 38, 45, 51, 58, 65, 73, 80, 89, 97, 107, 116, 126, 134, 146, 158, 169, 180, 192, 204, 218, 228, 243, 257, 270, 285, 302, 316, 331, 346, 364, 379, 397, 414, 433, 451, 468, 484, 505, 523, 544, 563, 584, 603, 625
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
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