A078184 Number of solutions to n^2 < x^2 + y^2 + z^2 < (n+1)^2; number of lattice points between spheres of radii n and n+1.

0, 20, 60, 128, 228, 380, 440, 684, 860, 1068, 1328, 1548, 1940, 2288, 2484, 2924, 3300, 3824, 4140, 4700, 5240, 5484, 6260, 6864, 7340, 8180, 8676, 9392, 9804, 10988, 11208, 12572, 13100, 13860, 14792, 15588, 16604, 17328, 18500, 19292

Offset

0,2

Links

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Formula

a(n) = A078183(n+1) - A000605(n).

Mathematica

Table[Sum[SquaresR[3, k], {k, n^2 + 1, (n + 1)^2 - 1}], {n, 0, 50}]

Prog

(Python)
from math import isqrt, prod
from sympy import factorint
def A078184(n): return ((n+1<<1)+sum(1+((s:=isqrt(t:=(n+1+k)*(n+1-k)))<<1)+(sum(isqrt(t-m**2) for m in range(s+1))<<2) for k in range(1, n+2))<<1)-6*prod(p**e+(0 if p&3==1 else (p**e-1)//(p-1)<<1) for p, e in factorint(n+1>>(~(n+1) & n).bit_length()).items())-((n<<1)+sum(1+((s:=isqrt(t:=(n+k)*(n-k)))<<1)+(sum(isqrt(t-m**2) for m in range(s+1))<<2) for k in range(1, n+1))<<1) # Chai Wah Wu, Feb 22 2026

Crossrefs

Cf. A000605, A078183.

Sequence in context: A344200 A163761 A154072 * A362268 A116530 A219830

Adjacent sequences: A078181 A078182 A078183 * A078185 A078186 A078187

Keyword

nonn

Author

T. D. Noe, Nov 21 2002

Status

approved