OFFSET
0,6
COMMENTS
A floretion-generated sequence: 'i + 0.5('ij' + 'ik' + 'ji' + 'jk' + 'ki' + 'kj').
From Greg Dresden, Nov 15 2024: (Start)
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:
._______________. ._______________.
| |_ | _| _| | | |_ |_ | _| |
|___|_|_|___|___| |___|___|_|_|___|
._______________. ._______________.
| |_ | |_ |_ | | |_ |_ | |_ |
|___|_|___|___|_| |___|___|_|___|_| (End)
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Creighton Dement, Online Floretion Multiplier.
Yüksel Soykan, A Study on Generalized Jacobsthal-Padovan Numbers, Earthline Journal of Mathematical Sciences (2020) Vol. 4, No. 2, 227-251.
Index entries for linear recurrences with constant coefficients, signature (0,1,2).
FORMULA
G.f.: x^2/(1-x^2-2*x^3).
a(n) = A052947(n-2). - R. J. Mathar, Nov 10 2009
a(n) = a(n-2) + 2*a(n-3). - Wesley Ivan Hurt, May 23 2023
From Greg Dresden, Nov 17 2024: (Start)
a(2*n+1) = 2*a(n)^2 + 2*a(n+1)*a(n+2).
a(3*n+1) = Sum_{i=1..n} a(3*i-2)*2^(n-i). (End)
MATHEMATICA
LinearRecurrence[{0, 1, 2}, {0, 0, 1}, 60] (* Vladimir Joseph Stephan Orlovsky, May 24 2011 *)
CoefficientList[Series[x^2/(1-x^2-2x^3), {x, 0, 50}], x] (* Harvey P. Dale, May 29 2021 *)
PROG
(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
(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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Creighton Dement, Apr 08 2009
STATUS
approved