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

1,7

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

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.

Empirical 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

Prog

(PARI) { for (n=1, 500, write("b060550.txt", n, " ", 2^(floor(n/3) + (n%3)%2 - 1) - 2^(floor((n + 3)/6) + (n%6==1) - 1)); ) } \\ Harry J. Smith, Jul 07 2009

Crossrefs

Cf. A060547, A060548, A008611, A008615.

Sequence in context: A098361 A050977 A053448 * A099206 A269223 A121341

Adjacent sequences: A060547 A060548 A060549 * A060551 A060552 A060553

Keyword

easy,nonn

Author

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

Status

approved