[seqfan] Brick sequences

[franktaw at netscape.net]

My basic brick sequence is A005169 (known there and in the literature 
as a "coin fountain").  This is the sequence where mirror images are 
considered different.  I am still trying to generate the sequence where 
mirror images are considered the same.  If a(n) is the sequence I am 
trying to compute, and b(n) is the number of symmetric coin fountains, 
then a(n) = (A005169(n) + b(n)) / 2; so finding the number of symmetric 
coin fountains is an equivalent problem.

Looking at A005169 there is a formula: "A005169(n) = f(n, 1), where 
f(n, p) = 0 if p > n, 1 if p = n, Sum(1 <= q <= p+1; f(n-p, q)) if p < 
n".  By inspection, this makes f(n,k) the number of compositions of n 
with first part k, under the constraint that a(i+1) <= a(i) + 1.  This 
characterizes the fountains of coins; shift each row right 1/2 space 
compared to the row under it, count the columns, and you get a 
composition with this property.  The triangle of f(n,k) was not in the 
OEIS; I have submitted it as A168396.

The row sums of A168396 appear to be A003116, which has a conjecture to 
that effect.  I would like to see (a) a generating function for 
A168396, and (b) a proof of the conjecture for A003116.  (While these 
are separate problems, it is likely that (a) would lead to (b).)

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I have submitted the sequence for the number of stable piles of bricks 
(mirror images distinct):

%S A168368 1,1,1,2,4,7
%N A168368 Number of stable piles of n bricks.
%C A168368 This is similar to various sequences with rows of coins 
(often
pennies). Each brick must be offset by 1/2 brick from any bricks under 
it.
However, a brick might only have a brick under one half, provided the 
pile is
stable. A definition of stability is provided in the Paterson paper.
%H A168368 Mike Paterson et al, <a 
href="http://math.dartmouth.edu/~pw/papers/maxover.pdf">Maximum
Overhang</a>
%e A168368 For n = 4, we have the following 4 piles:
%e A168368 ...........................|=|.
%e A168368 ...........|=|.......|=|..|=|=|
%e A168368 |=|=|=|=|.|=|=|=|.|=|=|=|..|=|.
%e A168368 The brick on top of the last pile stabilizes it.
%Y A168368 Cf. A005169, A001524.
%K A168368 bref,more,nonn
%O A168368 0,4

Obviously, one would like more terms, and ideally a program.

Franklin T. Adams-Watters