A060549 - OEIS

%I #15 Feb 01 2015 06:20:36

%S 0,1,2,2,6,6,12,14,28,28,60,60,120,124,248,248,504,504,1008,1016,2032,

%T 2032,4080,4080,8160,8176,16352,16352,32736,32736,65472,65504,131008,

%U 131008,262080,262080,524160,524224,1048448,1048448,2097024,2097024

%N a(n) is the number of distinct patterns (modulo geometric D3-operations) with strict median-reflective (palindrome) symmetry (i.e., having no other symmetry) 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="/A060549/b060549.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^ceiling(n/2) - 2^(floor((n+3)/6) + d(n)), with d(n)=1 if n mod 6=1 else d(n)=0.

%F a(n) = A060546(n) - A060548(n) = 2^A008619(n-1) - 2^A008615(n+1), for n >= 1.

%F Empirical g.f.: x^2*(2*x^4 + 2*x^3 + 2*x + 1) / ((2*x^2-1)*(2*x^6-1)). - _Colin Barker_, Aug 29 2013

%o (PARI) { for (n=1, 500, write("b060549.txt", n, " ", 2^ceil(n/2) - 2^(floor((n + 3)/6) + (n%6==1))); ) } \\ _Harry J. Smith_, Jul 07 2009

%Y Cf. A060546, A060548, A008619, A008615.

%K easy,nonn

%O 1,3

%A André Barbé (Andre.Barbe(AT)esat.kuleuven.ac.be), Apr 03 2001