A380832 Number of points in Z^4 of norm <= n where the sum of the four entries is even.

1, 1, 49, 169, 625, 1465, 3337, 5689, 10009, 15937, 24865, 35761, 51265, 69817, 94849, 124009, 161497, 204529, 260137, 320497, 394705, 478705, 577489, 687913, 819313, 960457, 1127785, 1309153, 1517161, 1742497, 2001505, 2273473, 2585905, 2920009, 3297337, 3700153, 4144105, 4618657, 5145865, 5703073

Offset

0,3

Comments

Points in Z^4 with even sum of entries forms the D_4 lattice. That is to say, the sequence is the "ball" pattern on D_4 lattice.

a(n) == 1 (mod 24).

Links

Table of n, a(n) for n=0..39.

Steven Lu, Python program

Example

a(2) = 49, because in the ball with radius 2, there is 1 point (0,0,0,0), 8 points similar to (0,0,0,2), 24 points similar to (0,0,1,1), and 16 points similar to (1,1,1,1).

Prog

(Python) # See Steven Lu's link
(PARI) a(n) = sum(x=-n, n, sum(y=-n, n, sum(z=-n, n, sum(t=-n, n, (((x+y+z+t) % 2)==0) && (x^2+y^2+z^2+t^2 <=n^2))))); \\ Michel Marcus, Feb 09 2025

Crossrefs

Cf. A055410.

Sequence in context: A147608 A258060 A039686 * A038628 A244695 A244179

Adjacent sequences: A380829 A380830 A380831 * A380833 A380834 A380835

Keyword

nonn,new

Author

Steven Lu, Feb 05 2025

Status

approved