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

Author

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

Keywords

• easy
• nice
• nonn

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).

Crossrefs

Cf. A008611, A008615.

Programs

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 *)

PARI

a(n)=if(n<1,0,2^((n+3)\6+(n%6==1)))

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)), where 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.

From Colin Barker, Aug 29 2013: (Start)

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). (End)

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