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A083093 Triangle formed by reading Pascal's triangle (A007318) mod 3. 34
1, 1, 1, 1, 2, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 2, 1, 1, 0, 0, 2, 0, 0, 1, 1, 1, 0, 2, 2, 0, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 1, 0, 0, 0, 0, 0, 0, 1, 2, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
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]

REFERENCES

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, G. Skordev, Linear cellular automata, finite automata and Pascal's triangle, Disc. Appl. Math. 66 (1996) 1-22

Lin Jiu, Christophe Vignat, On Binomial Identities in Arbitrary Bases, arXiv:1602.04149 [math.CO], 2016.

Y. Moshe, The density of 0's in recurrence double sequences, J. Number Theory, 103 (2003), 109-121.

Y. Moshe, The distribution of elements in automatic double sequences, Discr. Math., 297 (2005), 91-103.

Index entries for triangles and arrays related to Pascal's triangle

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 xrange(21): print [T(n, k) for k in xrange(n + 1)] # Indranil Ghosh, Jul 26 2017

CROSSREFS

Cf. A007318, A051638 (row sums), A090044, A047999, A034931, A034930, A008975, A034932, A062296, A006047.

Cf. A006996 (central terms), A173019, A206424, A227428.

Sequences based on the triangles formed by reading Pascal's triangle mod m: A047999 (m = 2), A083093 (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).

Sequence in context: A113045 A204179 A204244 * A293899 A015794 A011650

Adjacent sequences:  A083090 A083091 A083092 * A083094 A083095 A083096

KEYWORD

easy,nonn,tabl

AUTHOR

Benoit Cloitre, Apr 22 2003

STATUS

approved

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Last modified February 18 05:48 EST 2018. Contains 299298 sequences. (Running on oeis4.)