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

Author

Steven Lu, Feb 05 2025

Keywords

• nonn

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).

Examples

			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).
		

Links

• Steven Lu, Python program

Crossrefs

Cf. A055410.

Programs

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

Python

# See Steven Lu's link