A077958 Expansion of 1/(1-2*x^3).

1, 0, 0, 2, 0, 0, 4, 0, 0, 8, 0, 0, 16, 0, 0, 32, 0, 0, 64, 0, 0, 128, 0, 0, 256, 0, 0, 512, 0, 0, 1024, 0, 0, 2048, 0, 0, 4096, 0, 0, 8192, 0, 0, 16384, 0, 0, 32768, 0, 0, 65536, 0, 0, 131072, 0, 0, 262144, 0, 0, 524288, 0, 0, 1048576, 0, 0, 2097152, 0, 0, 4194304, 0, 0, 8388608, 0

Offset: 0

Author

N. J. A. Sloane, Nov 17 2002

Keywords

• nonn
• easy

Comments

a(n) is the number of L-tromino tilings of the n X 2 rectangle (see Exercise 2 in Grinberg). - Stefano Spezia, Nov 26 2019

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1002
• Darij Grinberg, Math 222: Enumerative Combinatorics, Fall 2019: Homework 1, Drexel University, Department of Mathematics, 2019.
• Index entries for linear recurrences with constant coefficients, signature (0,0,2).

Crossrefs

Cf. A006801, A162673, A163433, A329185.

Programs

Magma

R:=PowerSeriesRing(Integers(), 80); Coefficients(R!( 1/(1-2*x^3) )); // G. C. Greubel, Jun 23 2019

Mathematica

CoefficientList[Series[1/(1-2*x^3),{x,0,80}],x] (* Vladimir Joseph Stephan Orlovsky, Jan 30 2012 *)
Riffle[Riffle[2^Range[0,30],0],0,{3,-1,3}] (* Harvey P. Dale, Dec 18 2012 *)

PARI

Vec(1/(1-2*x^3)+O(x^80)) \\ Charles R Greathouse IV, Sep 27 2012

PARI

a(n)= !(n%3)<<(n\3) \\ Ruud H.G. van Tol, Mar 28 2025

Sage

(1/(1-2*x^3)).series(x, 80).coefficients(x, sparse=False) # G. C. Greubel, Jun 23 2019

Formula

From Stefano Spezia, Nov 26 2019: (Start)

a(n) = 2^(n/3) if 3 divides n, otherwise a(n) = 0 (see Exercise 2 in Grinberg).

E.g.f.: (1/3)*(exp(-(-2)^(1/3)*x) + exp(2^(1/3)*x) + exp((-1)^(2/3)*2^(1/3)*x)). (End)