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