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

0,2

Links

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

Formula

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

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

Prog

(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

Crossrefs

A 4-dimensional version of A016728.

Cf. A046895.

Sequence in context: A212863 A019560 A130811 * A232051 A247927 A247928

Adjacent sequences: A350737 A350738 A350739 * A350741 A350742 A350743

Keyword

nonn

Author

Jeongseop Lee, Jan 12 2022

Status

approved