OFFSET
1,1
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.
FORMULA
a(n) = 2^A008611(n-1) for n >= 1.
Sum_{n>=1} 1/a(n) = 4. - Amiram Eldar, Dec 10 2022
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 *)
PROG
(PARI) { for (n=1, 500, write("b060547.txt", n, " ", 2^(floor(n/3) + (n % 3) % 2)); ) } \\ Harry J. Smith, Jul 07 2009
(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
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
CROSSREFS
KEYWORD
easy,nice,nonn
AUTHOR
André Barbé (Andre.Barbe(AT)esat.kuleuven.ac.be), Apr 03 2001
EXTENSIONS
More terms from James A. Sellers, Apr 04 2001
Name replaced with given formula by Peter Luschny, Nov 26 2022
STATUS
approved