Search: id:A047999
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%I A047999
%S A047999 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,
%T A047999 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,
%U A047999 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
%N A047999 Sierpinski's triangle (or gasket): triangle, read by rows, formed by 
               reading Pascal's triangle mod 2.
%C A047999 Also triangle giving successive states of cellular automaton generated 
               by "Rule 60" and "Rule 102". - Hans Havermann (pxp(AT)rogers.com), 
               May 26 2002
%C A047999 Also triangle formed by reading triangle of Eulerian numbers (A08292) 
               mod 2. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Oct 02 2003
%C A047999 Self-inverse when regarded as an infinite lower triangular matrix over 
               GF(2).
%C A047999 Start with [1], repeatedly apply the map 0 -> [00/00], 1 -> [10/11] [Allouche 
               and Berthe]
%C A047999 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
%C A047999 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
%C A047999 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
%C A047999 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
%D A047999 R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), 
               no. 8, 697-712.
%D A047999 Y. Moshe, The distribution of elements in automatic double sequences, 
               Discr. Math., 297 (2005), 91-103.
%D A047999 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; Chapter 3.
%H A047999 T. D. Noe, <a href="b047999.txt">Rows n=0..50 of triangle, flattened</
               a>
%H A047999 J.-P. Allouche and V. Berthe, <a href="http://www.lri.fr/~allouche/">
               Triangle de Pascal, complexite et automates.</a>
%H A047999 J. Baer, <a href="http://ccins.camosun.bc.ca/~jbritton/blaise/bigblaise.html">
               Explore patterns in Pascal's Triangle</a>
%H A047999 A. Bogomolny, <a href="http://www.cut-the-knot.org/ctk/Sierpinski.shtml">
               Dot Patterns and Sierpinski Gasket</a>
%H A047999 S. Butkevich, <a href="http://www.math.ohio-state.edu/~btk/Pascal">Pascal 
               Triangle Applet</a>
%H A047999 B. Cherowitzo, <a href="http://www-math.cudenver.edu/~wcherowi/jcorn5.html">
               Pascal's Triangle using Clock Arithmetic, Part I</a>
%H A047999 B. Cherowitzo, <a href="http://www-math.cudenver.edu/~wcherowi/jcorn6.html">
               Pascal's Triangle using Clock Arithmetic, Part II</a>
%H A047999 A. Granville, <a href="http://www.cecm.sfu.ca/organics/papers/granville/
               support/pascalform.html">Pascal's Triangle Interface</a>
%H A047999 I. Kobayashi et al., <a href="http://www.ies.co.jp/math/java/misc/PascalTriangle/
               PascalTriangle.html">Pascal's Triangle</a>
%H A047999 Dr. Math, <a href="http://www.mathforum.org/dr.math/faq/formulas/faq/
               regpoly.html">Regular polygon formulas>Regular polygon formulas</
               a>
%H A047999 National Curve Bank, <a href="http://curvebank.calstatela.edu/sierpinski/
               sierpinski.htm">Sierpinski Triangles</a>
%H A047999 F. Richman, <a href="http://www.math.fau.edu/Richman/bintype.htm">Pascal's 
               triangle modulo n</a>
%H A047999 F. Richman, <a href="http://www.math.fau.edu/Richman/mla/pascal.htm">
               Pascal's triangle modulo n</a>
%H A047999 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               SierpinskiSieve.html">Sierpinski Sieve</a>
%H A047999 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Rule60.html">Rule 60</a>
%H A047999 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Rule102.html">Rule 102</a>
%H A047999 <a href="Sindx_Ce.html#cell">Index entries for sequences related to cellular 
               automata</a>
%H A047999 <a href="Sindx_Pas.html#Pascal">Index entries for triangles and arrays 
               related to Pascal's triangle</a>
%H A047999 <a href="Sindx_Si.html#sieve">Index entries for sequences generated by 
               sieves</a>
%F A047999 Sum_{k>=0} T(n, k) = A001316(n) = 2^A000120(n).
%e A047999 Triangle begins:
%e A047999 .1,
%e A047999 .1,1,
%e A047999 .1,0,1,
%e A047999 .1,1,1,1,
%e A047999 .1,0,0,0,1,
%e A047999 .1,1,0,0,1,1,
%e A047999 .1,0,1,0,1,0,1,
%e A047999 .1,1,1,1,1,1,1,1,
%e A047999 .1,0,0,0,0,0,0,0,1,
%e A047999 .1,1,0,0,0,0,0,0,1,1,
%e A047999 .1,0,1,0,0,0,0,0,1,0,1,
%e A047999 .1,1,1,1,0,0,0,0,1,1,1,1,
%e A047999 .1,0,0,0,1,0,0,0,1,0,0,0,1,
%e A047999 ....
%t A047999 Mod[ Flatten[ NestList[ Prepend[ #, 0] + Append[ #, 0] &, {1}, 13]], 
               2] (from Robert G. Wilson v May 26 2004)
%o A047999 Contribution from Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Oct 
               10 2009: (Start)
%o A047999 (PARI) \\ Recurrence for Pascal's triangle mod p, here p = 2.
%o A047999 p = 2; s=13; T=matrix(s,s); T[1,1]=1;
%o A047999 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 ));
%o A047999 for(n=1,s,for(k=1,n,print1(T[n,k],", "))) (End)
%Y A047999 Cf. A007318, A054431, A001317, A008292, A083093, A034931, A034930, A008975, 
               A034932.
%Y A047999 Sequence in context: A144093 A143200 A166282 this_sequence A054431 A164381 
               A106470
%Y A047999 Adjacent sequences: A047996 A047997 A047998 this_sequence A048000 A048001 
               A048002
%K A047999 nonn,tabl,easy,nice
%O A047999 0,1
%A A047999 N. J. A. Sloane (njas(AT)research.att.com).
%E A047999 Additional links from Lekraj Beedassy (blekraj(AT)yahoo.com), Jan 22 
               2004

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