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^(n-1).
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 25-27.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1000
Peter Kagey, Symmetric Brick Stacking, Mathematics Stack Exchange.
FORMULA
G.f.: (x + 2*x^3*M(x^2) + x^2*M(x^2))/((1-x^3*M(x^2))*(1-x^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(1-2*x^2-3*x^4 + O(x^3*x^n)))/(2*x^2)); Vec((x + 2*x*h + h)/((1-x*h)*(1-h)))} \\ Andrew Howroyd, Mar 12 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Peter Kagey, Oct 10 2018
EXTENSIONS
a(20)-a(40) from Andrew Howroyd, Oct 25 2018
STATUS
approved