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

1,3

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.

Links

M. F. Hasler, Table of n, a(n) for n = 1..1000.

Project Euler, Problem 86: Cuboid Route, (2005)

Formula

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

Example

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.

Prog

(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])

Crossrefs

Cf. A143714 (first differences).

Sequence in context: A101437 A039856 A301703 * A159685 A370804 A251729

Adjacent sequences: A143712 A143713 A143714 * A143716 A143717 A143718

Keyword

easy,nonn

Author

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

Status

approved