A060549 a(n) is the number of distinct patterns (modulo geometric D3-operations) with strict median-reflective (palindrome) symmetry (i.e., having no other 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. 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.

0, 1, 2, 2, 6, 6, 12, 14, 28, 28, 60, 60, 120, 124, 248, 248, 504, 504, 1008, 1016, 2032, 2032, 4080, 4080, 8160, 8176, 16352, 16352, 32736, 32736, 65472, 65504, 131008, 131008, 262080, 262080, 524160, 524224, 1048448, 1048448, 2097024, 2097024

Offset: 1

Author

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

Keywords

• easy
• nonn

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 (0,2,0,0,0,2,0,-4).

Crossrefs

Cf. A060546, A060548, A008619, A008615.

Programs

PARI

a(n) = { 2^ceil(n/2) - 2^(floor((n + 3)/6) + (n%6==1)) } \\ Harry J. Smith, Jul 07 2009

Formula

a(n) = 2^ceiling(n/2) - 2^(floor((n+3)/6) + d(n)), with d(n)=1 if n mod 6=1 else d(n)=0.

a(n) = A060546(n) - A060548(n) = 2^A008619(n-1) - 2^A008615(n+1), for n >= 1.

G.f.: x^2*(2*x^4 + 2*x^3 + 2*x + 1) / ((2*x^2-1)*(2*x^6-1)). - Colin Barker, Aug 29 2013