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A108279 a(n) = number of squares with corners on an n X n grid, distinct up to congruence. 6
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 (list; graph; refs; listen; history; text; internal format)
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 a (n X n)-grid (or geoboard). - Martin Renner, May 03 2011

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..1000

H. 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-Fran├žois 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

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Last modified November 21 17:44 EST 2017. Contains 295004 sequences.