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 A060548 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. 6
 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, 32, 64, 32, 64, 64, 64, 64, 128, 64, 128, 128, 128, 128, 256, 128, 256, 256, 256, 256, 512, 256, 512, 512, 512, 512, 1024, 512, 1024, 1024, 1024, 1024, 2048, 1024, 2048 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 LINKS Harry J. Smith, Table of n, a(n) for n=1,...,500 A. Barbé, Symmetric patterns in the cellular automaton that generates Pascal's triangle modulo 2, Discr. Appl. Math. 105(2000), 1-38. Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,2). FORMULA a(n) = 2^{floor[(n+3)/6]+d(n)}, with d(n)=1 if n mod 6=1, else d(n)=0. a(n) = a(n-2)a(n-3)/a(n-5), n>5. 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 MATHEMATICA a[n_] := a[n] = a[n-2]*a[n-3]/a[n-5]; a = a = a = a = 2; a = 1; Table[a[n], {n, 1, 63}] (* Jean-François Alcover, Dec 27 2011, after second formula *) LinearRecurrence[{0, 0, 0, 0, 0, 2}, {2, 1, 2, 2, 2, 2}, 70] (* Harvey P. Dale, Sep 19 2016 *) PROG (PARI) a(n)=if(n<1, 0, 2^((n+3)\6+(n%6==1))) (PARI) { for (n=1, 500, write("b060548.txt", n, " ", 2^((n + 3)\6 + (n%6==1))); ) } \\ Harry J. Smith, Jul 07 2009 CROSSREFS a(n) = 2^A008615(n+1) = 2^floor[A008611(n+2)/2] for n >= 1. Sequence in context: A010554 A062610 A025801 * A140426 A146879 A231577 Adjacent sequences:  A060545 A060546 A060547 * A060549 A060550 A060551 KEYWORD easy,nice,nonn AUTHOR André Barbé (Andre.Barbe(AT)esat.kuleuven.ac.be), Apr 02 2001 STATUS approved

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Last modified November 20 02:34 EST 2019. Contains 329323 sequences. (Running on oeis4.)