OFFSET
0,6
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]
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.
FORMULA
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) = (1/2)*Sum_{k = 0..n} mod(C(n,k)^2 - C(n,k), 3). - Peter Bala, Dec 17 2020
EXAMPLE
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
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 *)
PROG
(Haskell)
a227428 = sum . map (flip div 2) . a083093_row
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Jul 11 2013
STATUS
approved