A374512 - OEIS

%I #13 Jul 19 2024 14:39:14

%S 1,0,2,4,6,16,32,64,140,288,600,1264,2632,5504,11520,24064,50320,

%T 105216,219936,459840,961376,2009856,4201984,8784896,18366144,

%U 38397440,80275840,167829248,350873728,733556736,1533616128,3206266880,6703206656,14014111744

%N Number of ways to tile a 3 X n board with 2 X 2 and 3 X 3 staircase tiles.

%C Here are the 2 X 2 and 3 X 3 staircase tiles, both of which can be rotated as desired:

%C           _

%C    _     | |_

%C   | |_   |   |_

%C   |___|  |_____|.

%C This is a natural generalization of A127864, which counts the number of ways to tile a 2 X n board with 1 X 1 and 2 X 2 staircase tiles.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,4,2).

%F a(n) = 2*a(n-2) + 4*a(n-3) + 2*a(n-4).

%F a(2*n) = A108485(n).

%F a(2*n+3) = 4*Sum_{k=0..n} a(2*k)*A002605(n+1-k).

%F G.f.: 1/(1 - 2*x^2 - 4*x^3 - 2*x^4).

%e Here is one of the a(6)=32 ways to tile the 3 X 6 board:

%e    ___________

%e   | |_  |    _|

%e   |   |_|  _| |

%e   |_____|_|___|.

%t LinearRecurrence[{0, 2, 4, 2}, {1, 0, 2, 4}, 50]

%Y Cf. A002605, A127864, A108485.

%K nonn,easy

%O 0,3

%A _Greg Dresden_ and Shaolun Han, Jul 09 2024