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A108279 a(n) = number of squares with corners on an n X n grid, distinct up to congruence. 6


%S 0,1,3,5,8,11,15,18,23,28,33,38,45,51,58,65,73,80,89,97,107,116,126,

%T 134,146,158,169,180,192,204,218,228,243,257,270,285,302,316,331,346,

%U 364,379,397,414,433,451,468,484,505,523,544,563,584,603,625

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

%C 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.

%C 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

%H Andrew Howroyd, <a href="/A108279/b108279.txt">Table of n, a(n) for n = 1..1000</a>

%H H. Bottomley, <a href="/A002415/a002415.gif">Illustration of initial terms of A002415</a>

%e a(3)=3 because the 6 different squares that can be drawn on a 3 X 3 square lattice come in 3 sizes:

%e 4 squares of side length 1:

%e x.x.o....o.x.x....o.o.o....o.o.o

%e x.x.o....o.x.x....x.x.o....o.x.x

%e o.o.o....o.o.o....x.x.o....o.x.x

%e 1 square of side length sqrt(2):

%e o.x.o

%e x.o.x

%e o.x.o

%e 1 square of side length 2:

%e x.o.x

%e o.o.o

%e x.o.x

%e 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:

%e x.x.o.o....o.x.o.o....x.o.x.o....o.x.o.o....x.o.o.x

%e x.x.o.o....x.o.x.o....o.o.o.o....o.o.o.x....o.o.o.o

%e o.o.o.o....o.x.o.o....x.o.x.o....x.o.o.o....o.o.o.o

%e o.o.o.o....o.o.o.o....o.o.o.o....o.o.x.o....x.o.o.x

%t 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_ *)

%o (PARI)

%o 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

%Y Cf. A002415, A024206, A187452.

%K nonn

%O 1,3

%A _Hugo Pfoertner_, Jun 05 2005

%E More terms from _David W. Wilson_, Jun 07 2005

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Last modified February 18 09:38 EST 2019. Contains 320249 sequences. (Running on oeis4.)