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%I #28 Dec 10 2022 10:46:07
%S 2,1,2,2,2,2,4,2,4,4,4,4,8,4,8,8,8,8,16,8,16,16,16,16,32,16,32,32,32,
%T 32,64,32,64,64,64,64,128,64,128,128,128,128,256,128,256,256,256,256,
%U 512,256,512,512,512,512,1024,512,1024,1024,1024,1024,2048,1024,2048
%N a(n) is the number of D3-symmetric patterns that may be formed with a top-down 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="/A060548/b060548.txt">Table of n, a(n) for n=1,...,500</a>
%H André 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>.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,2).
%F a(n) = 2^A008615(n+1) = 2^floor[A008611(n+2)/2] for n >= 1.
%F a(n) = 2^{floor[(n+3)/6]+d(n)}, with d(n)=1 if n mod 6=1, else d(n)=0.
%F a(n) = a(n-2)a(n-3)/a(n-5), n>5.
%F Conjecture: a(n)=2*a(n-6) for n>1. G.f.: -x*(2*x^5+2*x^4+2*x^3+2*x^2+x+2) / (2*x^6-1). - _Colin Barker_, Aug 29 2013
%F Sum_{n>=1} 1/a(n) = 7. - _Amiram Eldar_, Dec 10 2022
%t a[n_] := a[n] = a[n-2]*a[n-3]/a[n-5]; a[1] = a[3] = a[4] = a[5] = 2; a[2] = 1; Table[a[n], {n, 1, 63}] (* _Jean-François Alcover_, Dec 27 2011, after second formula *)
%t LinearRecurrence[{0,0,0,0,0,2},{2,1,2,2,2,2},70] (* _Harvey P. Dale_, Sep 19 2016 *)
%o (PARI) a(n)=if(n<1,0,2^((n+3)\6+(n%6==1)))
%o (PARI) { for (n=1, 500, write("b060548.txt", n, " ", 2^((n + 3)\6 + (n%6==1))); ) } \\ _Harry J. Smith_, Jul 07 2009
%Y Cf. A008611, A008615.
%K easy,nice,nonn
%O 1,1
%A André Barbé (Andre.Barbe(AT)esat.kuleuven.ac.be), Apr 02 2001
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