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A183712 1/20 of the number of (n+1) X 3 0..4 arrays with every 2 X 2 subblock strictly increasing clockwise or counterclockwise with one decrease.

%I A183712 #41 Mar 02 2025 23:20:40
%S A183712 5,17,54,174,559,1797,5776,18566,59677,191821,616574,1981866,6370351,
%T A183712 20476345,65817520,211558554,680016837,2185791545,7025832918,
%U A183712 22583273462,72589861759,233327025821,749987665760,2410700161342,7748761123965,24906995867477
%N A183712 1/20 of the number of (n+1) X 3 0..4 arrays with every 2 X 2 subblock strictly increasing clockwise or counterclockwise with one decrease.
%C A183712 Column 2 of A183719. [Corrected by _M. F. Hasler_, Oct 07 2014]
%C A183712 This sequence counts closed walks of length (n+2) at the vertex of a triangle, to which a loop has been added to one of the remaining vertices and two loops has been added to the third vertex. - _David Neil McGrath_, Sep 04 2014
%C A183712 From _Greg Dresden_, Mar 02 2025: (Start)
%C A183712 a(n) is the number of ways to tile, with squares and dominoes, a 2 X n board with two extra spaces at the end. Here is the board for n=3:
%C A183712     _____
%C A183712    |_|_|_|_ _
%C A183712    |_|_|_|_|_|,
%C A183712 and here is one of the a(3)=54 possible tilings of this board:
%C A183712     _____
%C A183712    |___| |___
%C A183712    |_|_|_|___|.
%C A183712 Compare to A033505 (tilings of 2 X n board with one extra space at the end) and A030186 (tilings of 2 X n board with no extra spaces at the end). (End)
%H A183712 R. H. Hardin, <a href="/A183712/b183712.txt">Table of n, a(n) for n = 1..200</a>
%H A183712 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,1,-1).
%F A183712 a(n) = 3*a(n-1) + a(n-2) - a(n-3).
%F A183712 The top left element of A^(n+2) where A=(0,1,1;1,1,1;1,1,2). - _David Neil McGrath_, Sep 04 2014
%F A183712 a(n) ~ c*k^n where k = 1.629316... is the largest root of x^3 - 3x^2 - x + 1 and c = 1.6293... is conjecturally the largest root of 148x^3 - 296x^2 + 90x - 1. - _Charles R Greathouse IV_, Sep 15 2014
%F A183712 G.f.: x*(5+2*x-2*x^2) / (1-3*x-x^2+x^3). - _Colin Barker_, Mar 16 2016
%F A183712 a(n) = A030186(n) + A033505(n). - _Greg Dresden_, Mar 02 2025
%e A183712 Some solutions for 5 X 3:
%e A183712 ..0..1..4....1..2..0....4..0..4....4..3..4....4..0..4....1..4..0....3..4..2
%e A183712 ..3..2..3....0..3..4....2..1..3....0..2..0....3..2..3....2..3..1....1..0..1
%e A183712 ..4..1..0....1..2..1....4..0..4....4..3..4....0..1..0....0..4..0....2..4..3
%e A183712 ..3..2..3....0..3..4....3..2..3....0..2..1....4..2..3....1..3..1....1..0..1
%e A183712 ..4..0..4....1..2..1....4..1..0....4..3..0....0..1..0....0..4..0....2..3..2
%e A183712 ...
%e A183712 ...R..L.......R..L.......R..L.......L..R.......R..L.......L..R.......R..L...
%e A183712 ...L..R.......L..R.......L..R.......R..L.......L..R.......R..L.......L..R...
%e A183712 ...R..L.......R..L.......R..L.......L..R.......R..L.......L..R.......R..L...
%e A183712 ...L..R.......L..R.......L..R.......R..L.......L..R.......R..L.......L..R...
%o A183712 (PARI) a(n)=([0,1,1;1,1,1;1,1,2]^(n+2))[1,1] \\ _Charles R Greathouse IV_, Sep 15 2014
%o A183712 (PARI) Vec(x*(5+2*x-2*x^2)/(1-3*x-x^2+x^3) + O(x^50)) \\ _Colin Barker_, Mar 16 2016
%Y A183712 Cf. A183727, A183719, A030186, A033505.
%K A183712 nonn,walk,easy
%O A183712 1,1
%A A183712 _R. H. Hardin_, Jan 06 2011