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A143715 Number of subsets {a,b,c} of {1,...,n} such that (a+b)^2+c^2 is a square (where c = max(a,b,c)).

%I A143715 #11 Jul 12 2023 11:15:01
%S A143715 0,0,2,3,3,6,6,10,14,14,14,25,25,25,35,43,43,50,50,67,85,85,85,113,
%T A143715 113,113,123,139,139,158,158,173,191,191,197,230,230,230,244,286,286,
%U A143715 321,321,337,379,379,379,456,456,456,474,493,493,512,536,589,609,609,609
%N A143715 Number of subsets {a,b,c} of {1,...,n} such that (a+b)^2+c^2 is a square (where c = max(a,b,c)).
%C A143715 Also: Number of cuboids of side lengths not exceeding n such that the shortest path over the surface from one vertex to the opposite one is integral (cf. link to Project Euler).
%C A143715 Also: partial sums of A143714, i.e., number of triples (a,b,c), 1 <= a <= b <= c <= n, such that (a+b)^2+c^2 is a square.
%H A143715 M. F. Hasler, <a href="/A143715/b143715.txt">Table of n, a(n) for n = 1..1000</a>.
%H A143715 Project Euler, <a href="http://projecteuler.net/index.php?section=problems&amp;id=86">Problem 86: Cuboid Route</a>, (2005)
%F A143715 a(n) = sum( A143714(i), i=1..n ).
%e A143715 We have a(4) = a(5) = 3, corresponding to the cuboids of size 3 x 3 x 1, 3 x 2 x 2 and 4 x 2 x 1, i.e. to A143714(3)=2 and A143714(4)=1. No other cuboids with side lengths not exceeding 5 have the property that (a+b)^2+c^2 is a square. See A143714 for more details.
%o A143715 (PARI) A143715(M)=sum(a=1,M,sum(b=a,M,sum(c=b,M,issquare((a+b)^2+c^2))))
%o A143715 /* or: */ s=0; A143715=vector(100,i,s+=A143714[i])
%Y A143715 Cf. A143714 (first differences).
%K A143715 easy,nonn
%O A143715 1,3
%A A143715 _M. F. Hasler_, Aug 29 2008, Aug 30 2008