[seqfan] Brick sequences
franktaw at netscape.net
franktaw at netscape.net
Tue Nov 24 21:48:24 CET 2009
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).)
-----
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
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