A335274 a(n) = 2*a(n-1) + a(n-3), where a(0) = 0, a(1) = 1, a(2) = 4.

0, 1, 4, 8, 17, 38, 84, 185, 408, 900, 1985, 4378, 9656, 21297, 46972, 103600, 228497, 503966, 1111532, 2451561, 5407088, 11925708, 26302977, 58013042, 127951792, 282206561, 622426164, 1372804120, 3027814801, 6678055766, 14728915652, 32485646105, 71649347976

Offset

0,3

Comments

a(n) is the number of ways to tile a 2 x n strip, with a bent tromino added to the top, with dominos and L-shaped trominos:

_

|_|_

|_|_|_ _ _

|_|_|_|_|_| . . .

|_|_|_|_|_| . . .

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,0,1).

Formula

a(n) = 2*a(n-1) + a(n-3).

a(n) = 2*A008998(n-1) - A008998(n-4).

a(n) = A008998(n-1) + 2*A008998(n-2).

G.f.: x*(1 + 2*x) / (1 - 2*x - x^3). - Colin Barker, Jun 04 2020

Example

a(2) = 4 as shown by these four tilings:
   _         _         _         _
  |X|_      | |_      |X|_      | |_
  |X|X|  ,  |_|X|  ,  |X|X|  ,  |_| |
  |_ _|     |X X|     | | |     |X|_|
  |_ _|     |_ _|     |_|_|     |X X|

Mathematica

LinearRecurrence[{2, 0, 1}, {0, 1, 4}, 50] (* Paolo Xausa, Mar 20 2025 *)

Prog

(PARI) concat(0, Vec(x*(1 + 2*x) / (1 - 2*x - x^3) + O(x^35))) \\ Colin Barker, Jun 04 2020

Crossrefs

Cf. A008998, A335242.

Sequence in context: A019479 A084814 A098125 * A296399 A119471 A145779

Adjacent sequences: A335271 A335272 A335273 * A335275 A335276 A335277

Keyword

nonn,easy

Author

Michael Tulskikh, May 30 2020

Status

approved