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A077435 Number of right triangles whose vertices are lattice points in {1,2,...,n} X {1,2,...,n}. 11

%I #25 Mar 01 2017 11:03:43

%S 0,4,44,200,596,1444,2960,5520,9496,15332,23596,34936,50020,69732,

%T 94816,126176,164960,212372,269620,337960,418716,513444,623736,751152,

%U 897776,1065220,1255460,1470680,1713052,1984564,2288304,2626160,3000960,3415124,3871108

%N Number of right triangles whose vertices are lattice points in {1,2,...,n} X {1,2,...,n}.

%C It would be nice to have a formula. - _N. J. A. Sloane_, Jun 29 2016

%C Place all bounding boxes of A279433 that will fit into the n X n grid in all possible positions, and the proper rectangles in two orientations: a(n) = sum(i=1..n, sum(j=1..i, k * (n-i+1) * (n-j+1) * A279433(i,j) where k=1 when i=j and k=2 otherwise. - _Lars Blomberg_, Mar 01 2017

%H Lars Blomberg, <a href="/A077435/b077435.txt">Table of n, a(n) for n = 1..10000</a> (the first 184 terms from R. H. Hardin)

%e For n=2 if the four points are labeled

%e ab

%e cd

%e then the right triangles are abc, abd, acd, bcd,

%e so a(2)=4.

%e For n=3, label the points

%e abc

%e def

%e ghi

%e The right triangles are: abd (4*4 ways), acg (4 ways), acd and adf (8 ways each), ace and dbf (4 ways each), for a total of a(3) = 44. - _N. J. A. Sloane_, Jun 30 2016

%Y Cf. A187452, A279433.

%K nonn

%O 1,2

%A _John W. Layman_, Nov 30 2002

%E a(1) corrected by _Lars Blomberg_, Mar 01 2017

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