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A350834 Number of ways to tile an n X n right triangle with squares and dominoes, where vertical dominoes are only allowed in the largest vertical column.

%I A350834 #63 Jul 07 2022 16:35:40
%S A350834 1,1,3,11,73,749,12657,343693,15140923,1078147567,124268659473,
%T A350834 23172219304577,6991754237772409,3413365649747365697,
%U A350834 2696315730346059254139,3446235324323962173174283,7127008624714819485698797681,23848280807640171362927751869341
%N A350834 Number of ways to tile an n X n right triangle with squares and dominoes, where vertical dominoes are only allowed in the largest vertical column.
%C A350834 This is the third column in A229556.
%F A350834 a(n) = Fibonacci(n+1)*a(n-1) + Fibonacci(n)*Fibonacci(n-1)*a(n-2).
%F A350834 a(n) = P(n+1) + Sum_{i=2..n} a(i-1)*Fibonacci(i-1)*Fibonacci(n-(i-1))*P(n)/P(i), where P(n) = A003266(n) the product of the first n Fibonacci numbers. - _Greg Dresden_ and _Tianle Tina Yao_, Jul 03 2022
%e A350834 Here is one of the 73 tilings for the n=4 case. Note that vertical dominoes are only allowed in the "first" column.
%e A350834 .  _
%e A350834   | |_
%e A350834   |_|_|_
%e A350834   | |___|_
%e A350834   |_|_|___|
%t A350834 T[0] = 1; T[1] = 1; T[n_] := T[n] = Fibonacci[n + 1] T[n - 1] + (Fibonacci[n] Fibonacci[n - 1]) T[n - 2]; Table[T[n], {n, 0, 20}]
%Y A350834 Cf. A229556, A000045, A003266.
%K A350834 nonn
%O A350834 0,3
%A A350834 _Greg Dresden_ and _Tianle Tina Yao_, Jun 12 2022