A278761 a(n) is the number of parallelepipeds with vertices with integer coordinates between 0 and n and diagonals from one corner to the opposite corner with an integer length.

0, 0, 1, 1, 1, 1, 2, 4, 5, 6, 6, 7, 10, 10, 11, 12, 15, 16, 19, 20, 23, 25, 28, 30, 33, 35, 38, 40, 44, 47, 52, 54, 57, 59, 63, 65, 71, 73, 79, 81, 86, 89, 98, 101, 106, 108, 114, 117, 126, 130, 137, 142, 147, 150, 159, 162, 173, 178, 182

Offset

0,7

Links

Table of n, a(n) for n=0..58.

Formula

Sides a,b,c must satisfy the conditions that (1) a^2 + b^2 + c^2 = d^2 and (2) a, b, and c are positive, coprime, and not > n.

Example

n=8: [1,2,2],[1,4,8],[2,3,6],[4,4,7],[6,6,7].

Prog

(Python)
def coprime(k, m, n):
    while m:
        k, m=m, k%m
    if k==1:return 1
    while k:
        n, k=k, n%k
    return n
def a(n):
    kv=set(i**2 for i in range(2*n))
    pyt=0
    for a in range(1, n):
        for b in range(a, n+1):
            for c in range(b, n+1):
                if a**2+b**2+c**2 in kv and coprime(a, b, c)==1:
                    pyt+=1
    return pyt
print([a(n) for n in range(70)])

Crossrefs

Sequence in context: A086370 A083788 A058317 * A287076 A344587 A100006

Adjacent sequences: A278758 A278759 A278760 * A278762 A278763 A278764

Keyword

nonn

Author

Knut Ångström, Nov 27 2016

Extensions

Formula clarified by Harvey P. Dale, Dec 08 2017

Status

approved