A159287 - OEIS

%I #44 Nov 18 2024 02:54:11

%S 0,0,1,0,1,2,1,4,5,6,13,16,25,42,57,92,141,206,325,488,737,1138,1713,

%T 2612,3989,6038,9213,14016,21289,32442,49321,75020,114205,173662,

%U 264245,402072,611569,930562,1415713,2153700,3276837,4985126,7584237,11538800

%N Expansion of x^2/(1-x^2-2*x^3).

%C A floretion-generated sequence: 'i + 0.5('ij' + 'ik' + 'ji' + 'jk' + 'ki' + 'kj').

%C From _Greg Dresden_, Nov 15 2024: (Start)

%C a(n) is the number of ways to tile a 2 X (n+1) board with L-shaped trominos and S-shaped quadrominos, where the first tile must be an upright L. For example, here are the a(7)=4 ways to tile a 2 X 8 board:

%C   ._______________.   ._______________.

%C   | |_  |  _|  _| |   | |_  |_  |  _| |

%C   |___|_|_|___|___|   |___|___|_|_|___|

%C   ._______________.   ._______________.

%C   | |_  | |_  |_  |   | |_  |_  | |_  |

%C   |___|_|___|___|_|   |___|___|_|___|_| (End)

%H G. C. Greubel, <a href="/A159287/b159287.txt">Table of n, a(n) for n = 0..1000</a>

%H Creighton Dement, <a href="https://web.archive.org/web/20211129085739/http://fumba.eu/sitelayout/Floretion.php">Online Floretion Multiplier</a>.

%H Yüksel Soykan, <a href="https://doi.org/10.34198/ejms.4220.227251">A Study on Generalized Jacobsthal-Padovan Numbers</a>, Earthline Journal of Mathematical Sciences (2020) Vol. 4, No. 2, 227-251.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,2).

%F G.f.: x^2/(1-x^2-2*x^3).

%F a(n) = A052947(n-2). - _R. J. Mathar_, Nov 10 2009

%F a(n) = a(n-2) + 2*a(n-3). - _Wesley Ivan Hurt_, May 23 2023

%F From _Greg Dresden_, Nov 17 2024: (Start)

%F a(2*n+1) = 2*a(n)^2 + 2*a(n+1)*a(n+2).

%F a(3*n+1) = Sum_{i=1..n} a(3*i-2)*2^(n-i). (End)

%t LinearRecurrence[{0, 1, 2}, {0, 0, 1}, 60] (* _Vladimir Joseph Stephan Orlovsky_, May 24 2011 *)

%t CoefficientList[Series[x^2/(1-x^2-2x^3),{x,0,50}],x] (* _Harvey P. Dale_, May 29 2021 *)

%o (PARI) a(n)=([0,1,0; 0,0,1; 2,1,0]^n*[0;0;1])[1,1] \\ _Charles R Greathouse IV_, Oct 03 2016

%o (Magma) I:=[0,0,1]; [n le 3 select I[n] else Self(n-2) + 2*Self(n-3): n in [1..30]]; // _G. C. Greubel_, Jun 27 2018

%Y Cf. A159284, A159285, A159286, A159288.

%Y Essentially the same as A052947.

%K easy,nonn

%O 0,6

%A _Creighton Dement_, Apr 08 2009