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)).

0, 0, 2, 3, 3, 6, 6, 10, 14, 14, 14, 25, 25, 25, 35, 43, 43, 50, 50, 67, 85, 85, 85, 113, 113, 113, 123, 139, 139, 158, 158, 173, 191, 191, 197, 230, 230, 230, 244, 286, 286, 321, 321, 337, 379, 379, 379, 456, 456, 456, 474, 493, 493, 512, 536, 589, 609, 609, 609

Offset: 1

Author

M. F. Hasler, Aug 29 2008, Aug 30 2008

Keywords

• easy
• nonn

Comments

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).

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.

Examples

			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.
		

Links

• M. F. Hasler, Table of n, a(n) for n = 1..1000.
• Project Euler, Problem 86: Cuboid Route, (2005)

Crossrefs

Cf. A143714 (first differences).

Programs

PARI

A143715(M)=sum(a=1,M,sum(b=a,M,sum(c=b,M,issquare((a+b)^2+c^2))))
/* or: */ s=0; A143715=vector(100,i,s+=A143714[i])

Formula

a(n) = sum( A143714(i), i=1..n ).