A232970 Expansion of (1-3*x)/(1-5*x+3*x^2+x^3).

1, 2, 7, 28, 117, 494, 2091, 8856, 37513, 158906, 673135, 2851444, 12078909, 51167078, 216747219, 918155952, 3889371025, 16475640050, 69791931223, 295643364940, 1252365390981, 5305104928862, 22472785106427, 95196245354568, 403257766524697, 1708227311453354, 7236167012338111, 30652895360805796

Offset: 0

Author

N. J. A. Sloane, Dec 05 2013

Keywords

• nonn
• easy

Comments

For n > 2, a(n) is the number of tilings of (a 2 X (n+1) rectangle missing the top right and top left 1 X 1 cells) using 1 X 1 squares, dominoes and right trominoes. Compare with similar tiling sequences A001076 and A110679. - Greg Dresden and Yilin Zhu, Jul 10 2025

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• M. Dziemianczuk, Counting Lattice Paths With Four Types of Steps, Graphs and Combinatorics, September 2013, DOI 10.1007/s00373-013-1357-1.
• Hermann Stamm-Wilbrandt, 6 interlaced bisections
• Index entries for linear recurrences with constant coefficients, signature (5,-3,-1).

Crossrefs

Cf. A001076, A110679.

Programs

Magma

I:=[1,2,7]; [n le 3 select I[n] else 5*Self(n-1)- 3*Self(n-2)-Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 24 2017

Mathematica

LinearRecurrence[{5, -3, -1}, {1, 2, 7}, 30] (* Vincenzo Librandi, Jun 24 2017 *)
CoefficientList[Series[(1-3x)/(1-5x+3x^2+x^3),{x,0,30}],x] (* Harvey P. Dale, Oct 19 2024 *)

PARI

Vec((1-3*x)/(1-5*x+3*x^2+x^3) + O(x^30)) \\ Felix Fröhlich, Apr 15 2019

Sage

[(fibonacci(3*n+1) +1)/2 for n in (0..30)] # G. C. Greubel, Apr 19 2019

Formula

a(n) = 5*a(n-1) - 3*a(n-2) - a(n-3). - N. J. A. Sloane, Jun 23 2017

a(n) = (Fibonacci(3*n+1) + 1)/2 = Sum_{k=0..n} Fibonacci(3*k-1). - Ehren Metcalfe, Apr 15 2019

a(2*n) = A294262(2*n); a(2*n+1) = A254627(2*n+2). See "6 interlaced bisections" link. - Hermann Stamm-Wilbrandt, Apr 18 2019