A385826 a(n) = Sum_{k=0..n} (binomial(n, k) mod 7).

1, 2, 4, 8, 16, 18, 22, 2, 4, 8, 16, 32, 36, 44, 4, 8, 16, 32, 43, 58, 67, 8, 16, 32, 36, 72, 60, 92, 16, 32, 43, 72, 81, 120, 121, 18, 36, 58, 60, 120, 100, 144, 22, 44, 67, 92, 121, 144, 169, 2, 4, 8, 16, 32, 36, 44, 4, 8, 16, 32, 64, 72, 88, 8, 16, 32, 64, 86

Offset: 0

Author

Chai Wah Wu, Jul 09 2025

Keywords

• nonn

Comments

Sum of n-th row of Pascal's triangle mod 7, A095142.

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, A385825.

Programs

Python

from gmpy2 import digits
from sympy.abc import x
from sympy import Poly, rem
def A385826(n):
    k = (1,3,2,6,4,5)
    s = digits(n,7)
    t = [s.count(str(i)) for i in range(1,7)]
    G = 2**t[0]*(2*x+2)**t[2]*(x**2+2)**t[1]*(3*x**3+4)**t[5]*(2*x**4+x**3+2)**t[3]*(2*x**5+2*x+2)**t[4]
    c = Poly(rem(G,x**6-1),x).all_coeffs()[::-1]
    return int(sum(k[i]*c[i] for i in range(len(c)) if c[i]))