OFFSET
0,5
COMMENTS
Start with [1], repeatedly apply the map 0 -> [000/000/000], 1 -> [111/120/100], 2 -> [222/210/200]. - Philippe Deléham, Apr 16 2009
{T(n,k)} is a fractal gasket with fractal (Hausdorff) dimension log(A000217(3))/log(3) = log(6)/log(3) = 1.63092... (see Reiter reference). Replacing values greater than 1 with 1 produces a binary gasket with the same dimension (see Bondarenko reference). - Richard L. Ollerton, Dec 14 2021
REFERENCES
Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.
Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.
LINKS
Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
J.-P. Allouche, F. von Haeseler, H.-O. Peitgen, and G. Skordev, Linear cellular automata, finite automata and Pascal's triangle, Disc. Appl. Math. 66 (1996) 1-22.
Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see pp. 130-132.
Ilya Gutkovskiy, Illustrations (triangle formed by reading Pascal's triangle mod m)
Lin Jiu and Christophe Vignat, On Binomial Identities in Arbitrary Bases, arXiv:1602.04149 [math.CO], 2016.
Yossi Moshe, The density of 0's in recurrence double sequences, J. Number Theory, 103 (2003), 109-121.
Yossi Moshe, The distribution of elements in automatic double sequences, Discr. Math., 297 (2005), 91-103.
Ashley Melia Reiter, Determining the dimension of fractals generated by Pascal's triangle, Fibonacci Quarterly, 31(2), 1993, pp. 112-120.
FORMULA
T(i, j) = binomial(i, j) mod 3.
T(n+1,k) = (T(n,k) + T(n,k-1)) mod 3. - Reinhard Zumkeller, Jul 11 2013
T(n,k) = Product_{i>=0} binomial(n_i,k_i) mod 3, where n = Sum_{i>=0} n_i*3^i and k = Sum_{i>=0} k_i*3^i, 0<=n_i, k_i <=2 [Allouche et al.]. - R. J. Mathar, Jul 26 2017
EXAMPLE
. Rows 0 .. 3^3:
. 0: 1
. 1: 1 1
. 2: 1 2 1
. 3: 1 0 0 1
. 4: 1 1 0 1 1
. 5: 1 2 1 1 2 1
. 6: 1 0 0 2 0 0 1
. 7: 1 1 0 2 2 0 1 1
. 8: 1 2 1 2 1 2 1 2 1
. 9: 1 0 0 0 0 0 0 0 0 1
. 10: 1 1 0 0 0 0 0 0 0 1 1
. 11: 1 2 1 0 0 0 0 0 0 1 2 1
. 12: 1 0 0 1 0 0 0 0 0 1 0 0 1
. 13: 1 1 0 1 1 0 0 0 0 1 1 0 1 1
. 14: 1 2 1 1 2 1 0 0 0 1 2 1 1 2 1
. 15: 1 0 0 2 0 0 1 0 0 1 0 0 2 0 0 1
. 16: 1 1 0 2 2 0 1 1 0 1 1 0 2 2 0 1 1
. 17: 1 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 1
. 18: 1 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 1
. 19: 1 1 0 0 0 0 0 0 0 2 2 0 0 0 0 0 0 0 1 1
. 20: 1 2 1 0 0 0 0 0 0 2 1 2 0 0 0 0 0 0 1 2 1
. 21: 1 0 0 1 0 0 0 0 0 2 0 0 2 0 0 0 0 0 1 0 0 1
. 22: 1 1 0 1 1 0 0 0 0 2 2 0 2 2 0 0 0 0 1 1 0 1 1
. 23: 1 2 1 1 2 1 0 0 0 2 1 2 2 1 2 0 0 0 1 2 1 1 2 1
. 24: 1 0 0 2 0 0 1 0 0 2 0 0 1 0 0 2 0 0 1 0 0 2 0 0 1
. 25: 1 1 0 2 2 0 1 1 0 2 2 0 1 1 0 2 2 0 1 1 0 2 2 0 1 1
. 26: 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1
. 27: 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 .
- Reinhard Zumkeller, Jul 11 2013
MAPLE
A083093 := proc(n, k)
modp(binomial(n, k), 3) ;
end proc:
seq(seq(A083093(n, k), k=0..n), n=0..10) ; # R. J. Mathar, Jul 26 2017
MATHEMATICA
Mod[ Flatten[ Table[ Binomial[n, k], {n, 0, 13}, {k, 0, n}]], 3] (* Robert G. Wilson v, Jan 19 2004 *)
PROG
(Haskell)
a083093 n k = a083093_tabl !! n !! k
a083093_row n = a083093_tabl !! n
a083093_tabl = iterate
(\ws -> zipWith (\u v -> mod (u + v) 3) ([0] ++ ws) (ws ++ [0])) [1]
-- Reinhard Zumkeller, Jul 11 2013
(Magma) /* As triangle: */ [[Binomial(n, k) mod 3: k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Feb 15 2016
(Python)
from sympy import binomial
def T(n, k):
return binomial(n, k) % 3
for n in range(21): print([T(n, k) for k in range(n + 1)]) # Indranil Ghosh, Jul 26 2017
(Python)
from math import comb, isqrt
def A083093(n):
def f(m, k):
if m<3 and k<3: return comb(m, k)%3
c, a = divmod(m, 3)
d, b = divmod(k, 3)
return f(c, d)*f(a, b)%3
return f(r:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)), n-comb(r+1, 2)) # Chai Wah Wu, Apr 30 2025
CROSSREFS
Cf. A007318, A051638 (row sums), A090044, A047999, A034931, A034930, A008975, A034932, A062296, A006047.
Sequences based on the triangles formed by reading Pascal's triangle mod m: A047999 (m = 2), (this sequence) (m = 3), A034931 (m = 4), A095140 (m = 5), A095141 (m = 6), A095142 (m = 7), A034930(m = 8), A095143 (m = 9), A008975 (m = 10), A095144 (m = 11), A095145 (m = 12), A275198 (m = 14), A034932 (m = 16).
KEYWORD
AUTHOR
Benoit Cloitre, Apr 22 2003
STATUS
approved
