A060550 - OEIS

%I #25 Dec 24 2024 13:46:25

%S 0,0,0,1,0,1,2,1,2,6,2,6,12,6,12,28,12,28,56,28,56,120,56,120,240,120,

%T 240,496,240,496,992,496,992,2016,992,2016,4032,2016,4032,8128,4032,

%U 8128,16256,8128,16256,32640,16256,32640,65280,32640

%N a(n) is the number of distinct patterns (modulo geometric D_3-operations) with no other than strict 120-degree rotational 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.

%C 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="/A060550/b060550.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>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,2,0,0,2,0,0,-4).

%F a(n) = 2^(floor(n/3) + (n mod 3) mod 2 - 1) - 2^(floor((n+3)/6) + d(n)-1), with d(n)=1 if n mod 6=1, otherwise d(n)=0.

%F a(n) = (A060547(n) - A060548(n))/2.

%F a(n) = 2^(A008611(n-1) - 1) + 2^(A008615(n+1) - 1), for n >= 1.

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

%o (PARI) a(n) = { 2^(floor(n/3) + (n%3)%2 - 1) - 2^(floor((n + 3)/6) + (n%6==1) - 1) } \\ _Harry J. Smith_, Jul 07 2009

%Y Cf. A060547, A060548, A008611, A008615.

%K easy,nonn

%O 1,7

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