%I #42 Dec 10 2022 10:46:03

%S 2,1,2,4,2,4,8,4,8,16,8,16,32,16,32,64,32,64,128,64,128,256,128,256,

%T 512,256,512,1024,512,1024,2048,1024,2048,4096,2048,4096,8192,4096,

%U 8192,16384,8192,16384,32768,16384,32768,65536,32768,65536,131072

%N a(n) = 2^(floor(n/3) + ((n mod 3) mod 2)).

%C a(n) is the number of patterns, invariant under 120-degree rotations, that may appear in 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.

%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="/A060547/b060547.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>.

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

%F Sum_{n>=1} 1/a(n) = 4. - _Amiram Eldar_, Dec 10 2022

%p gf := (1+x^2+x^4)/(1-x^3)^2: s := series(gf, x, 100):

%p for i from 0 to 70 do printf(`%d,`,2^coeff(s,x,i)) od:

%p # Alternative:

%p a := n -> 2^(iquo(n, 3) + irem(irem(n, 3), 2));

%p seq(a(n), n = 1..49); # _Peter Luschny_, Nov 26 2022

%t CoefficientList[ Series[ (2x^2+x+2) / (1-2x^3), {x, 0, 48}], x] (* _Jean-François Alcover_, Nov 18 2011 *)

%o (PARI) { for (n=1, 500, write("b060547.txt", n, " ", 2^(floor(n/3) + (n % 3) % 2)); ) } \\ _Harry J. Smith_, Jul 07 2009

%o (Haskell)

%o a060547 = (2 ^) . a008611 . (subtract 1)

%o a060547_list = f [2,1,2] where f xs = xs ++ f (map (* 2) xs)

%o -- _Reinhard Zumkeller_, Nov 25 2013

%o def a_gen():

%o a, b, c = 1, 2, 4

%o yield b

%o while True:

%o yield a

%o a, b, c = b, c, a + a

%o a = a_gen()

%o print([next(a) for _ in range(51)]) # _Peter Luschny_, Nov 26 2022

%Y Cf. A008611, A060550.

%K easy,nice,nonn

%O 1,1

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

%E More terms from _James A. Sellers_, Apr 04 2001

%E Name replaced with given formula by _Peter Luschny_, Nov 26 2022