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

1,3

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

Cf. A002415, A024206, A187452.

Sequence in context: A207038 A081401 A003311 * A002821 A046992 A001463

Adjacent sequences: A108276 A108277 A108278 * A108280 A108281 A108282

Keyword

nonn

Author

Hugo Pfoertner, Jun 05 2005

Extensions

More terms from David W. Wilson, Jun 07 2005

Status

approved