A373882 Number of lattice points inside or on the 4-dimensional hypersphere x^2 + y^2 + z^2 + u^2 = 10^n.

9, 569, 49689, 4937225, 493490641, 49348095737, 4934805110729, 493480252693889, 49348022079085897, 4934802199975704129, 493480220066583590433, 49348022005552308828457, 4934802200546833521392241, 493480220054489318828539601, 49348022005446802425711456713, 4934802200544679211736756034457

Offset: 0

Author

Seiichi Manyama, Jun 21 2024

Keywords

• nonn

Links

• Wikipedia, Jacobi four square theorem

Crossrefs

Cf. A068785, A373881, A373883, A373884, A373885.

Cf. A024916, A046895.

Programs

PARI

b(k, n) = my(q='q+O('q^(n+1))); polcoef((eta(q^2)^5/(eta(q)^2*eta(q^4)^2))^k/(1-q), n);
a(n) = b(4, 10^n);

Python

from math import isqrt
def A373882(n): return 1+((-(s:=isqrt(a:=10**n))**2*(s+1)+sum((q:=a//k)*((k<<1)+q+1) for k in range(1,s+1))&-1)<<2)+(((t:=isqrt(m:=a>>2))**2*(t+1)-sum((q:=m//k)*((k<<1)+q+1) for k in range(1,t+1))&-1)<<4) # Chai Wah Wu, Jun 21 2024

Formula

a(n) = A046895(10^n).

a(n) == 1 (mod 8).

Limit_{n->oo} a(n) = Pi^2*100^n/2. - Hugo Pfoertner, Jun 21 2024