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A060552 a(n) is the number of distinct (modulo geometric D3-operations) nonsymmetric (no reflective nor rotational symmetry) patterns which can be formed by an equilateral triangular arrangement of closely packed black and white cells satisfying the local matching rule of Pascal's triangle modulo 2, where n is the number of cells in each edge of the arrangement. The matching rule is such that any elementary top-down triangle of three neighboring cells in the arrangement contains either one or three white cells. 1

%I

%S 0,0,0,1,2,7,14,35,70,154,310,650,1300,2666,5332,10788,21588,43428,

%T 86856,174244,348488,697992,1396040,2794120,5588240,11180680,22361360,

%U 44730896,89462032,178940432,357880864,715794960

%N a(n) is the number of distinct (modulo geometric D3-operations) nonsymmetric (no reflective nor rotational symmetry) patterns which can be formed by an equilateral triangular arrangement of closely packed black and white cells satisfying the local matching rule of Pascal's triangle modulo 2, where n is the number of cells in each edge of the arrangement. The matching rule is such that any elementary top-down triangle of three neighboring cells in the arrangement contains either one or three white cells.

%D A. Barb\'{e}, Symmetric patterns in the cellular automaton that generates Pascal's triangle modulo 2, Discr. Appl. Math. 105(2000), 1-38.

%H Harry J. Smith, <a href="/A060552/b060552.txt">Table of n, a(n) for n=1,...,500</a>

%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>

%F a(n)={2^(n-1)-2^[floor(n/3)+(n mod 3)mod 2-1]}/3+2^{floor[(n+3)/6]+d(n)-1} -2^floor[(n-1)/2], with d(n)=1 if n mod 6=1 else d(n)=0.

%F Contribution from _R. J. Mathar_, Aug 03 2009: (Start)

%F a(n)= 2*a(n-1) +2*a(n-2) -2*a(n-3) -4*a(n-4) -4*a(n-5) +10*a(n-6) -4*a(n-7) -4*a(n-8) +4*a(n-9) +8*a(n-10) +8*a(n-11) -16*a(n-12).

%F G.f.: -x^4*(-1-x^2-x^4+2*x^3+2*x^5+2*x^6)/((2*x-1)*(2*x^2-1)*(2*x^3-1)*(2*x^6-1)). (End)

%o (PARI) { for (n=1, 500, a=(2^(n-1)-2^(floor(n/3)+(n%3)%2-1))/3+2^(floor((n+3)/6)+(n%6==1)-1)-2^floor((n-1)/2); write("b060552.txt", n, " ", a); ) } [From _Harry J. Smith_, Jul 07 2009]

%Y A060552(n)=[A000079(n-1) - A060547(n)/2]/3 + A060548(n)/2 -A060546(n)/2 A060552(n)={A000079(n-1) - 2^[A008611(n-1)-1]}/3+ 2^[A008615(n+1)-1] -2^[A008619(n-1)-1], n >= 1

%K easy,nonn

%O 1,5

%A Andr\'{e} Barb\'{e} (Andre.Barbe(AT)esat.kuleuven.ac.be), Apr 03 2001

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Last modified December 20 09:43 EST 2014. Contains 252241 sequences.