%I #24 Sep 08 2022 08:45:08

%S 4,27,187,1302,9075,63267,441090,3075255,21440547,149482638,

%T 1042187067,7266087315,50658875658,353191693599,2462438631411,

%U 17168025532662,119694800484387,834507453158019,5818153224352338,40563936024707079,282810170576026755

%N 1/6 of the number of ways of 3-coloring a 4 X n grid.

%C Also the number of 3-colorings of the P_4 X P_n grid graph up to permutation of the colors. - _Andrew Howroyd_, Jun 26 2017

%D Michael S. Paterson (Warwick), personal communication.

%H Alois P. Heinz, <a href="/A078100/b078100.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 (9,-15,6).

%F See A078099 for formula.

%F G.f.: x*(9*x-4-4*x^2) / (6*x^3-15*x^2+9*x-1). - _Alois P. Heinz_, Mar 23 2009

%p a:= n-> (Matrix([[27, 4, 2/3]]). Matrix([[9, 1, 0], [ -15, 0, 1], [6, 0, 0]])^n)[1, 3]: seq(a(n), n=1..30); # _Alois P. Heinz_, Mar 23 2009

%t LinearRecurrence[{9, -15, 6}, {4, 27, 187}, 21] (* _Jean-François Alcover_, Feb 13 2016 *)

%o (Magma) I:=[4,27,187]; [n le 3 select I[n] else 9*Self(n-1)-15*Self(n-2)+6*Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Feb 13 2016

%Y Row 4 of (1/2)*A078099.

%Y Row 4 of A207997.

%K nonn,easy

%O 1,1

%A _N. J. A. Sloane_, Dec 05 2002

%E More terms from _Alois P. Heinz_, Mar 23 2009

%E Name clarified by _Andrew Howroyd_, Jun 26 2017