Written: 12/12/12 (i.e. Dec. 12, 2012)
(Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger and arxiv.org)
This is the case with the Bétréma - Penaud brilliant idea that one of us (DZ) talked about here, and realized that the same argument can be used to enumerate much more general towers. Then the other one of us (SBE) found lots of new deep results (see the output files below).
If you want to see the enumerating sequences, algebraic equations, recurrences, and asymptotic behavior
for half-pyramids, pyramids, and towers, when the building blocks are allowed to have sizes in any
subset of {1,2,3,4,5} with two or more elements,
the input file would produce the
output file.
If you want to see the enumerating sequences, algebraic equations, recurrences, and asymptotic behavior
for half-pyramids, pyramids, and towers, when the building blocks are single blocks of sizes
from 2 to 6, where all interfaces are allowed,
the input file would produce the
output file.
If you want to see the enumerating sequences, algebraic equations, recurrences, and asymptotic behavior
for half-pyramids, pyramids, and towers, when the building blocks are single blocks of sizes
from 2 to 6, where all interfaces are allowed, EXCEPT exact line-up (like in the original 3^{n} theorem case)
the input file would produce the
output file.