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A060547 a(n) = 2^(floor(n/3) + ((n mod 3) mod 2)).

%I A060547 #51 Jul 02 2025 16:02:01
%S A060547 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 A060547 512,256,512,1024,512,1024,2048,1024,2048,4096,2048,4096,8192,4096,
%U A060547 8192,16384,8192,16384,32768,16384,32768,65536,32768,65536,131072
%N A060547 a(n) = 2^(floor(n/3) + ((n mod 3) mod 2)).
%C A060547 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 A060547 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 A060547 Harry J. Smith, <a href="/A060547/b060547.txt">Table of n, a(n) for n = 1..500</a>
%H A060547 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 A060547 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>.
%H A060547 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0, 0, 2).
%F A060547 a(n) = 2^A008611(n-1) for n >= 1.
%F A060547 Sum_{n>=1} 1/a(n) = 4. - _Amiram Eldar_, Dec 10 2022
%p A060547 gf := (1+x^2+x^4)/(1-x^3)^2: s := series(gf, x, 100):
%p A060547 for i from 0 to 70 do printf(`%d,`,2^coeff(s,x,i)) od:
%p A060547 # Alternative:
%p A060547 a := n -> 2^(iquo(n, 3) + irem(irem(n, 3), 2));
%p A060547 seq(a(n), n = 1..49); # _Peter Luschny_, Nov 26 2022
%t A060547 CoefficientList[ Series[ (2x^2+x+2) / (1-2x^3), {x, 0, 48}], x] (* _Jean-François Alcover_, Nov 18 2011 *)
%o A060547 (PARI) a(n) = { 2^(floor(n/3) + (n % 3) % 2) } \\ _Harry J. Smith_, Jul 07 2009
%o A060547 (Haskell)
%o A060547 a060547 = (2 ^) . a008611 . (subtract 1)
%o A060547 a060547_list = f [2,1,2] where f xs = xs ++ f (map (* 2) xs)
%o A060547 -- _Reinhard Zumkeller_, Nov 25 2013
%o A060547 (Python)
%o A060547 def a_gen():
%o A060547     a, b, c = 1, 2, 4
%o A060547     yield b
%o A060547     while True:
%o A060547         yield a
%o A060547         a, b, c = b, c, a + a
%o A060547 a = a_gen()
%o A060547 print([next(a) for _ in range(51)]) # _Peter Luschny_, Nov 26 2022
%Y A060547 Cf. A008611, A060550.
%K A060547 easy,nice,nonn
%O A060547 1,1
%A A060547 André Barbé (Andre.Barbe(AT)esat.kuleuven.ac.be), Apr 03 2001
%E A060547 More terms from _James Sellers_, Apr 04 2001
%E A060547 Name replaced with given formula by _Peter Luschny_, Nov 26 2022