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

2, 1, 2, 4, 2, 4, 8, 4, 8, 16, 8, 16, 32, 16, 32, 64, 32, 64, 128, 64, 128, 256, 128, 256, 512, 256, 512, 1024, 512, 1024, 2048, 1024, 2048, 4096, 2048, 4096, 8192, 4096, 8192, 16384, 8192, 16384, 32768, 16384, 32768, 65536, 32768, 65536, 131072

Offset: 1

Author

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

Keywords

• easy
• nice
• nonn

Comments

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.

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.

Links

• Harry J. Smith, Table of n, a(n) for n = 1..500
• André Barbé, Symmetric patterns in the cellular automaton that generates Pascal's triangle modulo 2, Discr. Appl. Math. 105 (2000), 1-38.
• Index entries for sequences related to cellular automata.
• Index entries for linear recurrences with constant coefficients, signature (0, 0, 2).

Crossrefs

Cf. A008611, A060550.

Programs

Haskell

a060547 = (2 ^) . a008611 . (subtract 1)
a060547_list = f [2,1,2] where f xs = xs ++ f (map (* 2) xs)
-- Reinhard Zumkeller, Nov 25 2013

Maple

gf := (1+x^2+x^4)/(1-x^3)^2: s := series(gf, x, 100):
for i from 0 to 70 do printf(`%d,`,2^coeff(s,x,i)) od:
# Alternative:
a := n -> 2^(iquo(n, 3) + irem(irem(n, 3), 2));
seq(a(n), n = 1..49); # Peter Luschny, Nov 26 2022

Mathematica

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

PARI

a(n) = { 2^(floor(n/3) + (n % 3) % 2) } \\ Harry J. Smith, Jul 07 2009

Python

def a_gen():
    a, b, c = 1, 2, 4
    yield b
    while True:
        yield a
        a, b, c = b, c, a + a
a = a_gen()
print([next(a) for  in range(51)]) # _Peter Luschny, Nov 26 2022

Formula

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

Sum_{n>=1} 1/a(n) = 4. - Amiram Eldar, Dec 10 2022

Extensions

More terms from James Sellers, Apr 04 2001

Name replaced with given formula by Peter Luschny, Nov 26 2022