A060550 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.

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, 240, 496, 240, 496, 992, 496, 992, 2016, 992, 2016, 4032, 2016, 4032, 8128, 4032, 8128, 16256, 8128, 16256, 32640, 16256, 32640, 65280, 32640

Offset: 1

Author

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

Keywords

• easy
• nonn

Comments

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
• A. 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 (0,0,2,0,0,2,0,0,-4).

Crossrefs

Cf. A060547, A060548, A008611, A008615.

Programs

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

Formula

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.

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

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

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