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Number of first-quadrant cells (including the two boundaries) that are ON at stage n of the cellular automaton described in A079317.
6

%I #19 Jul 13 2023 17:26:37

%S 1,3,3,7,5,11,9,21,11,25,15,35,19,45,29,73,31,77,35,87,39,97,49,125,

%T 53,135,63,163,73,191,101,273,103,277,107,287,111,297,121,325,125,335,

%U 135,363,145,391,173,473,177,483,187,511,197,539,225,621,235,649,263,731

%N Number of first-quadrant cells (including the two boundaries) that are ON at stage n of the cellular automaton described in A079317.

%C Start with cell (0,0) active; at each succeeding stage the cells that share exactly one edge with an active cell change their state.

%C The pattern has 4-fold symmetry; sequence just counts cells in one quadrant.

%C This is not the CA discussed by Singmaster in the reference given in A079314. That was an error based on my misreading of the paper. - _N. J. A. Sloane_, Aug 05 2009

%D D. Singmaster, On the cellular automaton of Ulam and Warburton, M500 Magazine of the Open University, #195 (December 2003), pp. 2-7.

%H David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="/A000695/a000695_1.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]

%H D. Singmaster, <a href="/A079314/a079314.pdf">On the cellular automaton of Ulam and Warburton</a>, 2003 [Cached copy, included with permission]

%H N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>

%o (PARI) M=matrix(101,101); M[1,1]=1; for(s=1,100, c=[]; a=M[1,1]; for(x=2,100, for(y=2,100, a+=M[x,y]; if(M[x-1,y]+M[x+1,y]+M[x,y-1]+M[x,y+1]==1, c=concat(c,[[x,y]]) )); a+=M[x,1]+M[1,x]; if(M[x,2]==0 && M[x-1,1]+M[x+1,1]==1, c=concat(c,[[x,1]]) ); if(M[2,x]==0 && M[1,x-1]+M[1,x+1]==1, c=concat(c,[[1,x]]) )); print1(a,", "); for(i=1,length(c),M[c[i][1],c[i][2]]=1-M[c[i][1],c[i][2]]) ) - _Max Alekseyev_, Feb 02 2007

%Y Cf. A079317, A151922, A151923.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Feb 12 2003

%E More terms from _Max Alekseyev_, Feb 02 2007

%E Edited by _N. J. A. Sloane_, Aug 05 2009