A385741 a(n) = Sum_{k=0..n} (binomial(n, k) mod 9).

1, 2, 4, 8, 16, 14, 28, 38, 31, 8, 16, 32, 28, 56, 49, 62, 52, 68, 28, 56, 76, 62, 79, 122, 91, 92, 112, 8, 16, 32, 28, 56, 76, 80, 124, 140, 28, 56, 103, 80, 142, 158, 145, 146, 184, 62, 124, 158, 100, 146, 184, 188, 232, 230, 28, 56, 76, 80, 151, 158, 136, 236

Offset

0,2

Comments

Sum of n-th row of Pascal's triangle mod 9, A095143.

Links

Chai Wah Wu, Table of n, a(n) for n = 0..10000

James G. Huard, Blair K. Spearman, and Kenneth S. Williams, Pascal's triangle (mod 9), Acta Arithmetica, 78 (1997), 331-349.

Mathematica

a[n_]:=Sum[Mod[Binomial[n, k], 9], {k, 0, n}]; Table[a[n], {n, 0, 61}] (* James C. McMahon, Jul 10 2025 *)

Prog

(Python)
from gmpy2 import digits
import re, sympy
from sympy import S, I, sqrt, simplify, Rational
def A385741(n):
    s = digits(n, 3)
    n1 = s.count('1')
    n2 = s.count('2')
    n01 = s.count('10')
    n02 = s.count('20')
    n11 = len(re.findall('(?=11)', s))
    n12 = s.count('21')
    n121 = len(re.findall('(?=121)', s))
    n122 = s.count('221')
    n21 = s.count('12')
    n22 = len(re.findall('(?=22)', s))
    x1 = (3*(3**n2*(12*n01+(n02<<4)+3*n11+(n12<<2))-(n01+n12<<2)+(n02<<4)+n11)<<n1)>>3
    beta = S.Half*(I*sqrt(3)-1)
    def ind2(t): return (0, 0, 1, 0, 2, 5, 0, 4, 3)[t]
    def X(t): return beta**(ind2(t)-n11-n12+n121-n122)*(2-beta)**(n21-n121)*(3+beta)**(n2-n12-n21-n22+n121+n122)
    def Y(t): return beta**(n11-ind2(t))*(1-beta)**(n21-n121)*(2+beta)**(n2-n21-n22)*(1+2*beta)**n121
    def f(t): return ((3**n2<<n1)+((1<<n1) if (t+1)%3 else -(1<<n1))+((-1 if (not (t+1)%3)+n11+n122&1 else 1)<<n1-n11+n122+1)*sympy.re(X(t))+((not n122)*(-1 if n11&1 else 1)*3**(n22-n122)<<n1-n11+1)*sympy.re(Y(t)))*Rational(1, 6)
    x2 = simplify(sum(i*f(i) for i in (1, 2, 4, 5, 7, 8)))
    return int(x1+x2)
(PARI) a(n) = sum(k=0, n, binomial(n, k) % 9); \\ Michel Marcus, Jul 10 2025

Crossrefs

Cf. A001316, A051638, A384715, A385285.

Sequence in context: A334629 A358708 A108565 * A066005 A066600 A210025

Adjacent sequences: A385738 A385739 A385740 * A385742 A385743 A385744

Keyword

nonn

Author

Chai Wah Wu, Jul 09 2025

Status

approved