

A333650


Triangle read by rows: T(n,k) gives the number of domino towers of height k consisting of n bricks.


6



1, 1, 2, 1, 4, 4, 1, 7, 11, 8, 1, 12, 24, 28, 16, 1, 20, 52, 70, 68, 32, 1, 33, 110, 168, 193, 160, 64, 1, 54, 228, 401, 497, 510, 368, 128, 1, 88, 467, 944, 1257, 1412, 1304, 832, 256, 1, 143, 949, 2187, 3172, 3736, 3879, 3248, 1856, 512
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OFFSET

1,3


COMMENTS

The towers must have a contiguous base of bricks, and each brick must be at least half supported below by another brick. The stacks do not need to be stable.
Conjecture: For n > 1, T(n,2) = A000071(n+2).
A038622(n1,k) appears to give the number of domino towers consisting of n bricks with a base of k bricks.
The above conjectures can be summarized as follows:
T(2*n + k, n + k) = d_{n}(n + k  1) for k >= 1 and 0 <= n <= 3, where
d_{0}(m) = 2^(m1)*2;
d_{1}(m) = 2^(m3)*(10 + 6*m);
d_{2}(m) = 2^(m5)*(70 + 43*m + 9*m^2);
d_{3}(m) = 2^(m7)*(588 + 367*m + 84*m^2 + 9*m^3). (End)


REFERENCES

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, pages 2527.


LINKS



FORMULA

Row sums are given by A000244(n1) = 3^(n1).
T(n,1) = 1.
T(n,n) = 2^(n1).


EXAMPLE

Table begins:
n\k 1 2 3 4 5 6 7 8 9 10 11
+
1  1
2  1 2
3  1 4 4
4  1 7 11 8
5  1 12 24 28 16
6  1 20 52 70 68 32
7  1 33 110 168 193 160 64
8  1 54 228 401 497 510 368 128
9  1 88 467 944 1257 1412 1304 832 256
10  1 143 949 2187 3172 3736 3879 3248 1856 512
11  1 232 1916 5010 7946 9778 10766 10360 7920 4096 1024
.
T(3,2) = 4 because there are four domino towers of height two consisting of three bricks:
+++ ++ ++
      
+++++, +++++, +++++, and
       
++ +++ +++
++
 
+++++.
  
+++


CROSSREFS

Cf. A000071 (col. 2), A339493 (col. 3), A000244, A038622, A168368, A264746, A320314, A339252, A339254, A339029, A339346, A339494.


KEYWORD



AUTHOR



STATUS

approved



