cp's OEIS Frontend

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

%I A077435 #31 Apr 21 2025 03:20:21
%S A077435 0,4,44,200,596,1444,2960,5520,9496,15332,23596,34936,50020,69732,
%T A077435 94816,126176,164960,212372,269620,337960,418716,513444,623736,751152,
%U A077435 897776,1065220,1255460,1470680,1713052,1984564,2288304,2626160,3000960,3415124,3871108
%N A077435 Number of right triangles whose vertices are lattice points in {1,2,...,n} X {1,2,...,n}.
%C A077435 It would be nice to have a formula. - _N. J. A. Sloane_, Jun 29 2016
%H A077435 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)
%F A077435 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
%e A077435 For n=2 if the four points are labeled
%e A077435   ab
%e A077435   cd
%e A077435 then the right triangles are abc, abd, acd, bcd, so a(2)=4.
%e A077435 For n=3, label the points
%e A077435   abc
%e A077435   def
%e A077435   ghi
%e A077435 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 A077435 Cf. A187452, A279433.
%K A077435 nonn
%O A077435 1,2
%A A077435 _John W. Layman_, Nov 30 2002
%E A077435 a(1) corrected by _Lars Blomberg_, Mar 01 2017