A227428 Number of twos in row n of triangle A083093.

0, 0, 1, 0, 0, 2, 1, 2, 4, 0, 0, 2, 0, 0, 4, 2, 4, 8, 1, 2, 4, 2, 4, 8, 4, 8, 13, 0, 0, 2, 0, 0, 4, 2, 4, 8, 0, 0, 4, 0, 0, 8, 4, 8, 16, 2, 4, 8, 4, 8, 16, 8, 16, 26, 1, 2, 4, 2, 4, 8, 4, 8, 13, 2, 4, 8, 4, 8, 16, 8, 16, 26, 4, 8, 13, 8, 16, 26, 13, 26, 40

Offset: 0

Author

Reinhard Zumkeller, Jul 11 2013

Keywords

• nonn

Comments

"The number of entries with value r in the n-th row of Pascal's triangle modulo k is found to be 2^{#r^k (n)}, where now #_r^k (n) gives the number of occurrences of the digit r in the base-k representation of the integer n." [Wolfram] - _R. J. Mathar, Jul 26 2017 [This is not correct: there are entries in the sequence that are not powers of 2. - Antti Karttunen, Jul 26 2017]

Examples

			Example of Wilson's formula: a(26) = 13 = 2^(0-1)*(3^3-1) = 26/2, where A062756(26)=0, A081603(26)=3, 26=(222)_3. - _R. J. Mathar_, Jul 26 2017
		

Links

• Reinhard Zumkeller (terms 0..1000) & Antti Karttunen, Table of n, a(n) for n = 0..19683
• R. Garfield and H. S. Wilf, The distribution of the binomial coefficients modulo p, J. Numb. Theory 41 (1) (1992) 1-5.
• Marcus Jaiclin, et al. Pascal's Triangle, Mod 2,3,5
• D. L. Wells, Residue counts modulo three for the fibonacci triangle, Appl. Fib. Numbers, Proc. 6th Int Conf Fib. Numbers, Pullman, 1994 (1996) 521-536.
• Avery Wilson, Pascal's Triangle Modulo 3, Mathematics Spectrum, 47-2 - January 2015, pp. 72-75.
• S. Wolfram, Geometry of binomial coefficients, Am. Math. Monthly 91 (9) (1984) 566-571.

Crossrefs

Cf. A006047, A062296, A062756, A083093, A081603, A206424, A206428.

Programs

Haskell

a227428 = sum . map (flip div 2) . a083093_row

Maple

A227428 := proc(n)
    local a;
    a := 0 ;
    for k from 0 to n do
        if A083093(n,k) = 2 then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
seq(A227428(n),n=0..20) ; # R. J. Mathar, Jul 26 2017

Mathematica

Table[Count[Mod[Binomial[n, Range[0, n]], 3], 2], {n, 0, 99}] (* Alonso del Arte, Feb 07 2012 *)

PARI

A227428(n) = sum(k=0,n,2==(binomial(n,k)%3)); \\ (Naive implementation, from the description) Antti Karttunen, Jul 26 2017

Python

from sympy import binomial
def a(n):
    return sum(1 for k in range(n + 1) if binomial(n, k) % 3 == 2)
print([a(n) for n in range(101)]) # Indranil Ghosh, Jul 26 2017

Python

from sympy.ntheory import digits
def A227428(n):
    s = digits(n,3)[1:]
    return 3**s.count(2)-1<>1 # Chai Wah Wu, Jul 24 2025

Scheme

(define (A227428 n) (* (A000079 (- (A062756 n) 1)) (+ -1 (A000244 (A081603 n))))) ;; After Wilson's direct formula, Antti Karttunen, Jul 26 2017

Formula

a(n) = A006047(n) - A206424(n) = n + 1 - A062296(n) - A206424(n).

a(n) = 2^(N_1-1)*(3^N_2-1) where N_1 = A062756(n), N_2 = A081603(n). [Wilson, Theorem 2, Wells] - R. J. Mathar, Jul 26 2017

a(n) = A206424(n) * ((3^A081603(n))-1) / ((3^A081603(n))+1). - Antti Karttunen, Jul 27 2017

a(n) = (1/2)*Sum_{k = 0..n} mod(C(n,k)^2 - C(n,k), 3). - Peter Bala, Dec 17 2020