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A047999
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Sierpinski's triangle (or gasket): triangle, read by rows, formed by reading Pascal's triangle mod 2.
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93
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1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| Also triangle giving successive states of cellular automaton generated by "Rule 60" and "Rule 102". - Hans Havermann (gladhobo(AT)teksavvy.com), May 26 2002
Also triangle formed by reading triangle of Eulerian numbers (A08292) mod 2. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Oct 02 2003
Self-inverse when regarded as an infinite lower triangular matrix over GF(2).
Start with [1], repeatedly apply the map 0 -> [00/00], 1 -> [10/11] [Allouche and Berthe]
Also triangle formed by reading triangles A011117, A028338, A039757, A059438, A085881, A086646, A086872, A087903, A0104219 mod 2 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 18 2005
J. H. Conway writes (in Math Forum): at least the first 31 rows give odd-sided constructible polygons (sides 1, 3, 5, 15,17 ... see A001317). The 1's form a Sierpinski sieve. - M. Dauchez (mdzzdm(AT)yahoo.fr), Sep 19 2005
When regarded as an infinite lower triangular matrix, its inverse is a (-1,0,1)-matrix with zeros undisturbed and the nonzero entries in every column form the Prouhet-Thue-Morse sequence (1,-1,-1,1,-1,1,1,-1,...) A010060 (up to relabeling). - David Callan (callan(AT)stat.wisc.edu), Oct 27 2006
Triangle read by rows: antidiagonals of an array formed by successive iterates of running sums mod 2, beginning with (1, 1, 1,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 10 2008
T(n,k) = A057427(A143333(n,k)). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 24 2010]
The triangle sums, see A180662 for their definitions, link Sierpinski’s triangle A047999 with seven sequences, see the crossrefs. The Kn1y(n) and Kn2y(n), y >= 1, triangle sums lead to the Sierpinski-Stern triangle A191372. [From Johannes W. Meijer, Jun 5 2011]
Used to compute the total Steifel-Whitney cohomology class of the Real Projective space. This was an essential component of the proof that there are no product operations without zero divisors on R^n for n not equal to 1, 2, 4 or 8 (real numbers, complex numbers, quaternions, Cayley numbers), proven by Bott and Milnor. - Marcus Jaiclin, Feb 07 2012
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REFERENCES
| R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
Y. Moshe, The distribution of elements in automatic double sequences, Discr. Math., 297 (2005), 91-103.
H.-O. Peitgen, H. Juergens and D. Saupe: Chaos and Fractals (Springer-Verlag 1992), p. 408.
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; Chapter 3.
John W. Milnor and James D. Stasheff, Characteristic Classes, Princeton University Press, 1974, pp. 43-49 (sequence appears on pg. 46).
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LINKS
| T. D. Noe, Rows n=0..50 of triangle, flattened
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
J.-P. Allouche and V. Berthe, Triangle de Pascal, complexite et automates.
J. Baer, Explore patterns in Pascal's Triangle
A. Bogomolny, Dot Patterns and Sierpinski Gasket
S. Butkevich, Pascal Triangle Applet
B. Cherowitzo, Pascal's Triangle using Clock Arithmetic, Part I
B. Cherowitzo, Pascal's Triangle using Clock Arithmetic, Part II
A. Granville, Pascal's Triangle Interface
I. Kobayashi et al., Pascal's Triangle
Dr. Math, Regular polygon formulas>Regular polygon formulas
National Curve Bank, Sierpinski Triangles
F. Richman, Pascal's triangle modulo n
F. Richman, Pascal's triangle modulo n
Eric Weisstein's World of Mathematics, Sierpinski Sieve, Rule 60, Rule 102
Index entries for sequences related to cellular automata
Index entries for triangles and arrays related to Pascal's triangle
Index entries for sequences generated by sieves
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FORMULA
| Sum_{k>=0} T(n, k) = A001316(n) = 2^A000120(n).
T(n,k) = T(n-1,k-1) XOR T(n-1,k), 0<k<n; T(n,0) = T(n,n) = 1. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Dec 13 2009]
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EXAMPLE
| Triangle begins:
.1,
.1,1,
.1,0,1,
.1,1,1,1,
.1,0,0,0,1,
.1,1,0,0,1,1,
.1,0,1,0,1,0,1,
.1,1,1,1,1,1,1,1,
.1,0,0,0,0,0,0,0,1,
.1,1,0,0,0,0,0,0,1,1,
.1,0,1,0,0,0,0,0,1,0,1,
.1,1,1,1,0,0,0,0,1,1,1,1,
.1,0,0,0,1,0,0,0,1,0,0,0,1,
....
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MATHEMATICA
| Mod[ Flatten[ NestList[ Prepend[ #, 0] + Append[ #, 0] &, {1}, 13]], 2] (from Robert G. Wilson v May 26 2004)
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PROG
| Contribution from Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Oct 10 2009: (Start)
(PARI) \\ Recurrence for Pascal's triangle mod p, here p = 2.
p = 2; s=13; T=matrix(s, s); T[1, 1]=1;
for(n=2, s, T[n, 1]=1; for(k=2, n, T[n, k] = (T[n-1, k-1] + T[n-1, k])%p ));
for(n=1, s, for(k=1, n, print1(T[n, k], ", "))) (End)
(Haskell)
import Data.Bits (xor)
a047999 :: Int -> Int -> Int
a047999 n k = a047999_tabl !! n !! k
a047999_row n = a047999_tabl !! n
a047999_tabl = iterate (\row -> zipWith xor ([0] ++ row) (row ++ [0])) [1]
-- Reinhard Zumkeller, Dec 11 2011, Oct 24 2010
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CROSSREFS
| Cf. A007318, A054431, A001317, A008292, A083093, A034931, A034930, A008975, A034932.
From Johannes W. Meijer, Jun 5 2011: (Start)
A106344 is a skew version of this triangle.
Triangle sums (see the comments): A001316 (Row1; Related to Row2), A002487 (Related to Kn11, Kn12, Kn13, Kn21, Kn22, Kn23), A007306 (Kn3, Kn4), A060632 (Fi1, Fi2), A120562 (Ca1, Ca2), A112970 (Gi1, Gi2), A127830 (Ze3, Ze4). (End)
Sequence in context: A144093 A143200 A166282 * A054431 A164381 A106470
Adjacent sequences: A047996 A047997 A047998 * A048000 A048001 A048002
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KEYWORD
| nonn,tabl,easy,nice,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Additional links from Lekraj Beedassy (blekraj(AT)yahoo.com), Jan 22 2004
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