A385825 -id:A385825 - cp's OEIS frontend

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

Chai Wah Wu, Jul 09 2025

nonn

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

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.

Cf. A001316, A051638, A384715, A385285, A385741, A385825.

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]))

A385828 a(n) = Sum_{k=0..n} (binomial(n, k) mod 11).

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 31, 40, 36, 50, 56, 2, 4, 8, 16, 32, 64, 62, 80, 72, 100, 112, 4, 8, 16, 32, 53, 106, 91, 116, 100, 156, 169, 8, 16, 32, 64, 62, 124, 116, 188, 134, 224, 228, 16, 32, 53, 62, 91, 182, 144, 222, 202, 272, 291, 32, 64, 106, 124, 182, 188, 244

Offset: 0

Chai Wah Wu, Jul 09 2025

nonn

Sum of n-th row of Pascal's triangle mod 11, A095144.

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.

Cf. A001316, A051638, A384715, A385285, A385741, A385825, A385826.

from gmpy2 import digits
from sympy.abc import x
from sympy import Poly, rem
def A385828(n):
    s = digits(n,11)
    t = tuple(s.count(digits(i,11)) for i in range(1,11))
    G = 2**t[0]*(x+2)**t[1]*(5*x**5+6)**t[9]*(2*x**8+2)**t[2]*(2*x**5+2*x**4+2)**t[4]*(x**9+2*x**2+2)**t[3]*(2*x**7+2*x**5+2*x+2)**t[6]*(2*x**9+2*x**3+x**2+4)**t[7]*(2*x**9+x**6+2*x**2+2)**t[5]*(2*x**8+2*x**7+2*x**6+2*x**4+2)**t[8]
    c = Poly(rem(G,x**10-1),x).all_coeffs()[::-1]
    return int(sum(pow(2,i,11)*c[i] for i in range(len(c)) if c[i]))

cp's OEIS Frontend

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

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