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A350740
Number of integer points (x, y, z, w) at distance <= 1/2 from 3-sphere of radius n.
1
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
KEYWORD
nonn
AUTHOR
Jeongseop Lee, Jan 12 2022
STATUS
approved