OFFSET
1,2
COMMENTS
A domino tower is a stack of bricks, where (1) the bottom row is contiguous, and (2) each brick is supported from below by at least half of a brick. Note, that in this definition of domino towers, rows need not be offset by half a brick. The number of domino towers with n bricks is given by 4^(n-1).
In this sequence we want all rows to be convex, rather than just the bottom row.
LINKS
T. M. Brown, Convex domino towers, J. of Integer Sequences, 20 (2017), p.17.
FORMULA
G.f.: G(x) := [ Sum_{l>0} z^l (z^3 T(3,l)+(2 z^2-1) T(2,l)+(2 z+1) T(1,l)) ] / (z^5 T(2,3)+(3 z-1) z^3 T(1,3)+(4 z^3-3 (z+1) z+1) T(1,2)) , where
T(i,j) := A(i)B(j)-A(j)B(i),
A(l) := Sum_{n>=0} (z^(l n+n^2+n) (-z;z)_n)/((z;z)_n)^2,
B(l) := Sum_{n>=0} (z^(l n+n^2+n) (-z;z)_n)/((z;z)_n)^2 * (l+n+Sum_{m=1,...,n} (3 z^m+1)/(1-z^(2 m))), and
(a;q)_n is the q-Pochhammer symbol
EXAMPLE
For n=2, the a(2) = 4 domino towers are:
+-------+-------+
| | |
+-------+-------+
+-------+
| |
+---+---+---+
| |
+-------+
+-------+
| |
+-------+
| |
+-------+
+-------+
| |
+---+---+---+
| |
+-------+
For n=4, the 4^(n-1)-a(n)=64-61=3 domino towers, which have non-convex rows are:
+-------+ +-------+
| | | |
+-------+---+---+---+
| | |
+-------+-------+
+-------+ +-------+
| | | |
+---+---+---+-------+
| | |
+-------+-------+
+-------+ +-------+
| | | |
+---+---+---+---+---+---+
| | |
+-------+-------+
MATHEMATICA
f[n_, l_] := (f[n, l] =
Sum[(3 - 2 i + 2 l) f[n - i, i], {i, 1, Min[n, l + 1]}]);
f[0, l_] := 1;
Table[Sum[f[n - l, l], {l, 1, n}], {n, 1, 20}]
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Alexander M. Haupt, Nov 01 2020
STATUS
approved