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Number of connectable 3 X n binary matrices.
2

%I #20 Sep 08 2022 08:45:01

%S 1,3,13,87,585,3899,25973,173039,1152849,7680691,51171485,340922567,

%T 2271346969,15132518507,100818201477,671686589663,4475014115745,

%U 29814130048611,198632300941357,1323358787022391,8816685256575721

%N Number of connectable 3 X n binary matrices.

%C A connected (0,1) matrix is one where you can get from any black square, i.e., a1, to any other by chess king moves. A matrix is connectable if it is not connected, has rightmost column [1,0,1]' and becomes connected when any of [1,1,1]', [1,1,0]', [0,1,1]' or [0,1,0]' is appended.

%D R. Levy and J. Shapiro, Uniqueness in paint-by-numbers puzzles, preprint, 2000.

%H Vincenzo Librandi, <a href="/A054420/b054420.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-3,5).

%F a(n) = 7*a(n-1) - 3*a(n-2) + 5*a(n-3).

%F G.f.: x*(1+x)*(1-5*x)/(1 - 7*x + 3*x^2 - 5*x^3). - _Colin Barker_, Mar 08 2012

%t CoefficientList[Series[(1+x)*(1-5*x)/(1-7*x+3*x^2-5*x^3),{x,0,30}],x] (* _Vincenzo Librandi_, Apr 28 2012 *)

%o (Magma) I:=[1, 3, 13]; [n le 3 select I[n] else 7*Self(n-1)-3*Self(n-2)+5*Self(n-3): n in [1..25]]; // _Vincenzo Librandi_, Apr 28 2012

%Y Cf. A054417-A054421. A054420(n) = A054421(n-1) + 2*A054418(n-1).

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_, May 22 2000

%E More terms from _James A. Sellers_, May 23 2000