A060548 - OEIS

%I #36 Jan 07 2025 01:59:33

%S 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,

%T 32,64,32,64,64,64,64,128,64,128,128,128,128,256,128,256,256,256,256,

%U 512,256,512,512,512,512,1024,512,1024,1024,1024,1024,2048,1024,2048

%N 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.

%H Harry J. Smith, <a href="/A060548/b060548.txt">Table of n, a(n) for n = 1..500</a>

%H André Barbé, <a href="http://dx.doi.org/10.1016/S0166-218X(00)00211-0">Symmetric patterns in the cellular automaton that generates Pascal's triangle modulo 2</a>, Discr. Appl. Math. 105 (2000), 1-38.

%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,2).

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

%F a(n) = 2^(floor((n+3)/6) + d(n)), where d(n)=1 if n mod 6=1, else d(n)=0.

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

%F From _Colin Barker_, Aug 29 2013: (Start)

%F a(n) = 2*a(n-6) for n>1.

%F G.f.: -x*(2*x^5+2*x^4+2*x^3+2*x^2+x+2) / (2*x^6-1). (End)

%F Sum_{n>=1} 1/a(n) = 7. - _Amiram Eldar_, Dec 10 2022

%t 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 *)

%t LinearRecurrence[{0,0,0,0,0,2},{2,1,2,2,2,2},70] (* _Harvey P. Dale_, Sep 19 2016 *)

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

%Y Cf. A008611, A008615.

%K easy,nice,nonn

%O 1,1

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