

A320314


a(n) is the number of symmetric domino towers with n bricks.


3



1, 1, 3, 3, 7, 9, 19, 25, 53, 71, 149, 203, 423, 583, 1209, 1681, 3473, 4863, 10017, 14107, 28987, 41019, 84113, 119513, 244645, 348829, 712987, 1019731, 2081547, 2985097, 6086375, 8749185, 17820657, 25671983, 52241825, 75402907, 153316715, 221673707, 450393329, 652234089
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OFFSET

1,3


COMMENTS

A domino tower is a stack of bricks, where (1) each row is offset from the preceding row by half of a brick, (2) the bottom row is contiguous, and (3) each brick is supported from below by at least half of a brick.
The number of (not necessarily symmetric) domino towers with n blocks is given by 3^(n1).
a(n) is odd for all n.
The not necessarily symmetric case is described in the Miklos Bona reference. Similar considerations lead to a decomposition of symmetric towers into half pyramids which are enumerated by the Motzkin numbers.  Andrew Howroyd, Mar 12 2021


REFERENCES

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


LINKS



FORMULA

G.f.: (x + 2*x^3*M(x^2) + x^2*M(x^2))/((1x^3*M(x^2))*(1x^2*M(x^2))) where M(x) is the g.f. of A001006.  Andrew Howroyd, Mar 12 2021


EXAMPLE

For n = 4, the a(4) = 3 symmetric stacks are
++
 
+++++
  
+++++,
 
++
++ ++
   
+++++++, and
  
+++
+++++
    
+++++.


PROG

(PARI) seq(n)={my(h=(1  x^2  sqrt(12*x^23*x^4 + O(x^3*x^n)))/(2*x^2)); Vec((x + 2*x*h + h)/((1x*h)*(1h)))} \\ Andrew Howroyd, Mar 12 2021


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



