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%I #13 Feb 01 2015 03:51:36
%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.
%H Harry J. Smith, <a href="/A060552/b060552.txt">Table of n, a(n) for n=1..500</a>
%H A. Barbé, <a href="http://dx.doi.org/10.1016/S0166-218X(00)00211-0">Symmetric patterns in the cellular automaton that generates Pascal's triangle modulo 2</a>, Discr. Appl. Math. 105(2000), 1-38.
%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 a(n) = (A000079(n-1) - A060547(n)/2)/3 + A060548(n)/2 -A060546(n)/2.
%F a(n) = (A000079(n-1) - 2^(A008611(n-1) - 1))/3 + 2^(A008615(n+1) - 1) - 2^(A008619(n-1) - 1), n >= 1.
%F 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); ) } \\ _Harry J. Smith_, Jul 07 2009
%Y Cf. A000079, A060547, A060548, A060546, A060552, A008611, A008615, A008619.
%K easy,nonn
%O 1,5
%A André Barbé (Andre.Barbe(AT)esat.kuleuven.ac.be), Apr 03 2001
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