A078183 Number of solutions to x^2 + y^2 + z^2 < n^2; number of lattice points inside a sphere of radius n.

0, 1, 27, 93, 251, 485, 895, 1365, 2103, 2969, 4139, 5497, 7123, 9093, 11459, 13997, 17071, 20377, 24303, 28545, 33371, 38641, 44395, 50733, 57747, 65117, 73447, 82201, 91911, 101769, 112931, 124289, 137059, 150165, 164415, 179309, 195167

Offset

0,3

Links

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

Formula

a(n) = A000605(n) - A016725(n)

Mathematica

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

Prog

(Python)
from math import prod, isqrt
from sympy import factorint
def A078183(n): return 1+((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)-6*prod(p**e+(0 if p&3==1 else (p**e-1)//(p-1)<<1) for p, e in factorint(n>>(~n & n-1).bit_length()).items()) if n else 0 # Chai Wah Wu, Feb 22 2026

Crossrefs

Cf. A000605, A016725.

Sequence in context: A286192 A101100 A226359 * A072252 A154041 A233156

Adjacent sequences: A078180 A078181 A078182 * A078184 A078185 A078186

Keyword

nonn

Author

T. D. Noe, Nov 21 2002

Status

approved