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 A083093 Triangle, read by rows, 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 {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 B. A. Bondarenko, Generalized Pascal Triangles and Pyramids, Santa Clara, Calif.: The Fibonacci Association, 1993, pp. 130-132. 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. 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. 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. A. M. 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 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), (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). Sequence in context: A309365 A204179 A204244 * A334621 A293899 A015794 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 June 29 09:21 EDT 2022. Contains 354910 sequences. (Running on oeis4.)