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A374512 Number of ways to tile a 3 X n board with 2 X 2 and 3 X 3 staircase tiles.

%I A374512 #13 Jul 19 2024 14:39:14
%S A374512 1,0,2,4,6,16,32,64,140,288,600,1264,2632,5504,11520,24064,50320,
%T A374512 105216,219936,459840,961376,2009856,4201984,8784896,18366144,
%U A374512 38397440,80275840,167829248,350873728,733556736,1533616128,3206266880,6703206656,14014111744
%N A374512 Number of ways to tile a 3 X n board with 2 X 2 and 3 X 3 staircase tiles.
%C A374512 Here are the 2 X 2 and 3 X 3 staircase tiles, both of which can be rotated as desired:
%C A374512           _
%C A374512    _     | |_
%C A374512   | |_   |   |_
%C A374512   |___|  |_____|.
%C A374512 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 A374512 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,4,2).
%F A374512 a(n) = 2*a(n-2) + 4*a(n-3) + 2*a(n-4).
%F A374512 a(2*n) = A108485(n).
%F A374512 a(2*n+3) = 4*Sum_{k=0..n} a(2*k)*A002605(n+1-k).
%F A374512 G.f.: 1/(1 - 2*x^2 - 4*x^3 - 2*x^4).
%e A374512 Here is one of the a(6)=32 ways to tile the 3 X 6 board:
%e A374512    ___________
%e A374512   | |_  |    _|
%e A374512   |   |_|  _| |
%e A374512   |_____|_|___|.
%t A374512 LinearRecurrence[{0, 2, 4, 2}, {1, 0, 2, 4}, 50]
%Y A374512 Cf. A002605, A127864, A108485.
%K A374512 nonn,easy
%O A374512 0,3
%A A374512 _Greg Dresden_ and Shaolun Han, Jul 09 2024