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A159287
Expansion of x^2/(1-x^2-2*x^3).
7
0, 0, 1, 0, 1, 2, 1, 4, 5, 6, 13, 16, 25, 42, 57, 92, 141, 206, 325, 488, 737, 1138, 1713, 2612, 3989, 6038, 9213, 14016, 21289, 32442, 49321, 75020, 114205, 173662, 264245, 402072, 611569, 930562, 1415713, 2153700, 3276837, 4985126, 7584237, 11538800
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
Creighton Dement, Online Floretion Multiplier.
YĆ¼ksel Soykan, A Study on Generalized Jacobsthal-Padovan Numbers, Earthline Journal of Mathematical Sciences (2020) Vol. 4, No. 2, 227-251.
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
Essentially the same as A052947.
Sequence in context: A351253 A110332 A052947 * A252448 A021992 A337123
KEYWORD
easy,nonn,changed
AUTHOR
Creighton Dement, Apr 08 2009
STATUS
approved