A060551 a(n) is the number of nonsymmetric patterns (no reflective, nor rotational symmetry) which may 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, 0, 0, 6, 12, 42, 84, 210, 420, 924, 1860, 3900, 7800, 15996, 31992, 64728, 129528, 260568, 521136, 1045464, 2090928, 4187952, 8376240, 16764720, 33529440, 67084080, 134168160, 268385376, 536772192, 1073642592, 2147285184, 4294769760, 8589539520, 17179472064

Offset

1,4

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 (2,2,-2,-4,-4,10,-4,-4,4,8,8,-16).

Formula

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

a(n) = A000079(n) - 3*A060546(n) - A060547(n) + 3*A060548(n).

a(n) = A000079(n) - 3*2^A008619(n-1) - 2^A008611(n-1) + 3*2^A008615(n+1), for n >= 1.

G.f.: -6*x^4*(2*x^6 + 2*x^5 - x^4 + 2*x^3 - x^2 - 1) / ((2*x-1)*(2*x^2-1)*(2*x^3-1)*(2*x^6-1)). - Colin Barker, Aug 29 2013

Mathematica

LinearRecurrence[{2, 2, -2, -4, -4, 10, -4, -4, 4, 8, 8, -16}, {0, 0, 0, 6, 12, 42, 84, 210, 420, 924, 1860, 3900}, 40] (* Harvey P. Dale, Feb 01 2015 *)

Prog

(PARI) { for (n=1, 500, a=2^n-3*2^ceil(n/2)-2^(floor(n/3)+(n%3)%2)+3*2^(floor((n+3)/6)+(n%6==1)); write("b060551.txt", n, " ", a); ) } \\ Harry J. Smith, Jul 07 2009

Crossrefs

Cf. A000079, A060546, A060547, A060548, A008619, A008611, A008615.

Sequence in context: A206039 A048069 A152787 * A129113 A377119 A048025

Adjacent sequences: A060548 A060549 A060550 * A060552 A060553 A060554

Keyword

easy,nonn

Author

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

Extensions

More terms from Colin Barker, Aug 29 2013

Status

approved