A385825 a(n) = Sum_{k=0..n} (binomial(n, k) mod 5).

1, 2, 4, 8, 11, 2, 4, 8, 16, 22, 4, 8, 16, 22, 34, 8, 16, 22, 44, 48, 11, 22, 34, 48, 61, 2, 4, 8, 16, 22, 4, 8, 16, 32, 44, 8, 16, 32, 44, 68, 16, 32, 44, 88, 96, 22, 44, 68, 96, 122, 4, 8, 16, 22, 34, 8, 16, 32, 44, 68, 16, 32, 59, 68, 106, 22, 44, 68, 116, 142

Offset: 0

Author

Chai Wah Wu, Jul 09 2025

Keywords

• nonn

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..10000
• Richard Garfield and Herbert S. Wilf, The distribution of the binomial coefficients modulo p, Journal of Number Theory, 41 (1) (1992), 1-5.

Crossrefs

Cf. A001316, A051638, A384715, A385285, A385741.

Programs

Mathematica

a[n_]:=Sum[Mod[Binomial[n,k],5],{k,0,n}];Array[a,60,0] (* James C. McMahon, Jul 10 2025 *)

Python

from gmpy2 import digits
from sympy.abc import x
from sympy import Poly, rem
def A385825(n):
    k = (1,2,4,3)
    s = digits(n,5)
    t = [s.count(str(i)) for i in range(1,5)]
    G = 2**t[0]*(x+2)**t[1]*(2*x**2+3)**t[3]*(2*x**3+2)**t[2]
    c = Poly(rem(G,x**4-1),x).all_coeffs()[::-1]
    return int(sum(k[i]*c[i] for i in range(len(c)) if c[i]))

Formula

Row sums of A095140. - R. J. Mathar, Jul 19 2025