A108279 a(n) = number of squares with corners on an n X n grid, distinct up to congruence.

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

Author

Hugo Pfoertner, Jun 05 2005

Keywords

• nonn

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

Examples

			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
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000
• Henry Bottomley, Illustration of initial terms of A002415

Crossrefs

Cf. A002415, A024206, A187452.

Programs

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 *)

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

Extensions

More terms from David W. Wilson, Jun 07 2005