A143714 Number of pairs (a,b), 1 <= a <= b <= n, such that (a+b)^2+n^2 is a square.

0, 0, 2, 1, 0, 3, 0, 4, 4, 0, 0, 11, 0, 0, 10, 8, 0, 7, 0, 17, 18, 0, 0, 28, 0, 0, 10, 16, 0, 19, 0, 15, 18, 0, 6, 33, 0, 0, 14, 42, 0, 35, 0, 16, 42, 0, 0, 77, 0, 0, 18, 19, 0, 19, 24, 53, 20, 0, 0, 120, 0, 0, 60, 29, 30, 34, 0, 25, 24, 12, 0, 114, 0, 0, 46, 28, 18, 27, 0, 103, 28, 0, 0, 140

Offset: 1

Author

M. F. Hasler, Aug 29 2008

Keywords

• easy
• nonn

Comments

Number of cuboids of maximal side length n having integral shortest path going on the surface from one vertex to the opposite one.

Number of subsets {a,b} of {1..n} such that (a+b,n) form the shorter two legs of a Pythagorean triple.

For all primes p, p > 3: a(p)=0 (this directly follows from Sierpiński's proof that one of the shorter sides of a Pythagorean triple must be a multiple of 3, and one must be a multiple of 4). - Michael Turniansky, Jul 27 2016

Examples

			For n=3, we have the 3 X 3 X 1 and the 3 X 2 X 2 cuboid, for which the shortest path on the surface from one vertex to the opposite is of integral length sqrt(3^2 + (2+2)^2) = sqrt(3^2 + (3+1)^2) = 5.
For n=4, there is the 4 X 2 X 1 cuboid having this property.
For n=1,2 and 5 there is no cuboid having this property, i.e., no s >= 2, s <= 2n such that s^2 + n^2 would be a square.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..4096
• Project Euler, Problem 86: Cuboid route

Crossrefs

Cf. A143715 (partial sums).

Programs

PARI

A143714(M)=sum(a=1,M,sum(b=a,M,issquare((a+b)^2+M^2)))