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%I #7 Dec 08 2020 02:31:49
%S 1,2,1,5,3,1,5,9,4,1,3,14,14,5,1,1,16,29,20,6,1,0,12,46,51,27,7,1,0,5,
%T 52,101,81,35,8,1,0,1,41,150,190,120,44,9,1,0,0,22,169,345,323,169,54,
%U 10,1,0,0,7,143,495,687,511,229,65,11,1
%N T(n, k) is the number of domino towers of n bricks with height at most 3 and k bricks in the base floor. Triangle read by rows, T(n, k) for 1 <= k <= n.
%C This is the third triangle in a sequence of triangles: The first is the unit triangle A023531; the second is the binomial triangle C(k, n-k) without the first column, triangle A030528. This triangle highlights the connection between the Pascal triangle and the Fibonacci numbers in the case m = 2. Similarly, the current triangle and its row sums generalizes this to the case m = 3 of the construction of Union(A333650(n, j), j=1..m), classified by the number of bricks in the base floor.
%H Peter Luschny, <a href="/A339494/a339494.jpg">Domino towers classified by the size of the base floor</a>. Illustrating row 4 of the triangle.
%e Triangle starts: n: [row] sum
%e 1: [1] 1
%e 2: [2, 1] 3
%e 3: [5, 3, 1] 9
%e 4: [5, 9, 4, 1] 19
%e 5: [3, 14, 14, 5, 1] 37
%e 6: [1, 16, 29, 20, 6, 1] 73
%e 7: [0, 12, 46, 51, 27, 7, 1] 144
%e 8: [0, 5, 52, 101, 81, 35, 8, 1] 283
%e 9: [0, 1, 41, 150, 190, 120, 44, 9, 1] 556
%e 10: [0, 0, 22, 169, 345, 323, 169, 54, 10, 1] 1093
%Y Cf. A339495 (row sums), A333650, A030528, A023531.
%K nonn,tabl
%O 1,2
%A _Peter Luschny_, Dec 07 2020