login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations (which often say "discovered thanks to the OEIS").
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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 (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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 30 06:11 EST 2020. Contains 338781 sequences. (Running on oeis4.)