A060548 a(n) is the number of D3-symmetric patterns that may be formed with 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.

2, 1, 2, 2, 2, 2, 4, 2, 4, 4, 4, 4, 8, 4, 8, 8, 8, 8, 16, 8, 16, 16, 16, 16, 32, 16, 32, 32, 32, 32, 64, 32, 64, 64, 64, 64, 128, 64, 128, 128, 128, 128, 256, 128, 256, 256, 256, 256, 512, 256, 512, 512, 512, 512, 1024, 512, 1024, 1024, 1024, 1024, 2048, 1024, 2048

Offset

1,1

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.

Index entries for sequences related to cellular automata.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,2).

Formula

a(n) = 2^A008615(n+1) = 2^floor[A008611(n+2)/2] for n >= 1.

a(n) = 2^{floor[(n+3)/6]+d(n)}, with d(n)=1 if n mod 6=1, else d(n)=0.

a(n) = a(n-2)a(n-3)/a(n-5), n>5.

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

Sum_{n>=1} 1/a(n) = 7. - Amiram Eldar, Dec 10 2022

Mathematica

a[n_] := a[n] = a[n-2]*a[n-3]/a[n-5]; a[1] = a[3] = a[4] = a[5] = 2; a[2] = 1; Table[a[n], {n, 1, 63}] (* Jean-François Alcover, Dec 27 2011, after second formula *)
LinearRecurrence[{0, 0, 0, 0, 0, 2}, {2, 1, 2, 2, 2, 2}, 70] (* Harvey P. Dale, Sep 19 2016 *)

Prog

(PARI) a(n)=if(n<1, 0, 2^((n+3)\6+(n%6==1)))
(PARI) { for (n=1, 500, write("b060548.txt", n, " ", 2^((n + 3)\6 + (n%6==1))); ) } \\ Harry J. Smith, Jul 07 2009

Crossrefs

Cf. A008611, A008615.

Sequence in context: A062610 A351593 A025801 * A140426 A146879 A231577

Adjacent sequences: A060545 A060546 A060547 * A060549 A060550 A060551

Keyword

easy,nice,nonn

Author

André Barbé (Andre.Barbe(AT)esat.kuleuven.ac.be), Apr 02 2001

Status

approved