A350740 Number of integer points (x, y, z, w) at distance <= 1/2 from 3-sphere of radius n.

1, 32, 200, 528, 1280, 2744, 4272, 6592, 10144, 15048, 19824, 25824, 34744, 43520, 55184, 64680, 80864, 99184, 115616, 135144, 157344, 185872, 207304, 239600, 272960, 310240, 351096, 385392, 433040, 485528, 531728, 583696, 646056, 714800, 779488, 842928

Offset: 0

Author

Jeongseop Lee, Jan 12 2022

Keywords

• nonn

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..10000 (terms 0..200 from Robert Israel)

Crossrefs

A 4-dimensional version of A016728.

Cf. A046895.

Programs

Maple

N:= 40: # for a(0)..a(N)
V:= Array(0..N):
for x from 0 to N do
  for y from x to N do
    for z from y to N do
      for w from z to N do
        S:= {x,y,z,w};
        L:= [x,y,z,w];
        m:= round(sqrt(x^2 + y^2 + z^2 + w^2));
        if m > N then next fi;
        f:= 4!/mul(numboccur(s,L)!, s = S) * 2^(4 - numboccur(0,[x,y,z,w]));
        V[m]:= V[m] + f;
od od od od;
convert(V,list); # Robert Israel, Mar 08 2024

Python

from itertools import product
for R in range(100):
    c = 0
    for s in product(range(2*R + 1), repeat = 4):
        if (2*R - 1)**2 <= 4*sum((i - R)**2 for i in s) <= (2*R + 1)**2: c += 1
    print(c if R != 0 else 1, end = ', ')

Python

from itertools import combinations_with_replacement
from math import prod
from collections import Counter
def A350740(n):
    if n == 0: return 1
    x, y = (2*n-1)**2, (2*n+1)**2
    return sum(24//prod((1,1,2,6,24)[d] for d in q.values())<<4-q[0] for q in map(Counter,combinations_with_replacement(range(n+1),4)) if x <= sum(b*a**2 for a, b in q.items())<<2 <= y) # Chai Wah Wu, Jun 20 2024

Python

# Uses Python code in A046895
def A350740(n): return A046895(n*(n+1))-A046895(n*(n-1)) if n else 1 # Chai Wah Wu, Jun 21 2024

Formula

a(n) = A046895(n^2+n)-A046895(n^2-n) for n > 0. - Chai Wah Wu, Jun 21 2024