10 OEIS sequence.; [made Community Wiki]

I would like to count the polyominoes of $n$ squares that satisfy several restrictions:

1. Each is convex: every horizontal, or vertical line, meets the shape in either a single segment, or not at all.
2. The shape is tree-like in the sense that never does a $2 \times 2$ subarrangement of squares occur in the shape. So the dual graph is a tree.
3. I want to count only the free polyominoes, in that, to quote Wikipedia, "none is a rigid transformation (translation, rotation, reflection or glide reflection) of another."

Edit

Edit1 (25Jul11). My hand-accounting was repeatedly in error, corrected by Theo, Timothy, Gerhard, and Joel. As suggested by Gerhard, I now have a preliminary computer implementation, which shows that the number of polyominoes up to $n{=}8$ is 1,1,2,4,10,21,49,104, a sequence not (yet!) in the OEIS ("Sorry, but the terms do not match anything in the table.")

Edit

Edit2 (29Jul11). I now added the 104 octominoes I found, and I think I will stop there. Timothy Chow and Gerhard Paseman both analyzed the asymptotic growth as $2^{n-1}$ $\pm$ lesser terms, which accords with the (little) data I have. Certainly the count is going up by a bit more than a factor of 2 for each increase in $n$. It seems feasible through their analyses to obtain an exact count, although I have not given that a serious attempt. I thank everyone for their time & attention!
2:
4:
10:
21:
49:
104

Edit3 (19Jan12). At Neil Sloane's urging, I have submitted the sequence to OEIS. It is now A204804: Free tree-like convex polyominoes.

$n=3$, #=2:
$n=4$, #=4:
$n=5$, #=10:
$n=6$, #=21:
$n=7$, #=49:
$n=8$, #=104:

I would like to count the polyominoes of $n$ squares that satisfy several restrictions:

1. Each is convex: every horizontal, or vertical line, meets the shape in either a single segment, or not at all.
2. The shape is tree-like in the sense that never does a $2 \times 2$ subarrangement of squares occur in the shape. So the dual graph is a tree.
3. I want to count only the free polyominoes, in that, to quote Wikipedia, "none is a rigid transformation (translation, rotation, reflection or glide reflection) of another."

Edited

Edit (25Jul11). My hand-accounting was repeatedly in error, corrected by Theo, Timothy, Gerhard, and Joel. As suggested by Gerhard, I now have a preliminary computer implementation, which shows that the number of polyominoes up to $n{=}7$ heptominoes n{=}8$is 1,1,2,4,10,21,491,1,2,4,10,21,49,104, a sequence not (yet!) in the OEIS ("Sorry, but the terms do not match anything in the table.") Edit (29Jul11). I apologize for now added the inadequacy of 104 octominoes I found, and I think I will stop there. Timothy Chow and Gerhard Paseman both analyzed the following displaysasymptotic growth as$2^{n-1}\pm$lesser terms, which accords with the (little) data I will improve eventually: have. Certainly the count is going up by a bit more than a factor of 2 for each increase in$n$. It seems feasible through their analyses to obtain an exact count, although I have not given that a serious attempt. I thank everyone for their time & attention! 2: 4: 10: 21: 49: 104: 8 Hand accounting replaced by a computer enumeration. I would like to count the polyominoes of$n$squares that satisfy several restrictions: 1. Each is convex: every horizontal, or vertical line, meets the shape in either a single segment, or not at all. 2. The shape is tree-like in the sense that never does a$2 \times 2$subarrangement of squares occur in the shape. So the dual graph is a tree. 3. I want to count only the free polyominoes, in that, to quote Wikipedia, "none is a rigid transformation (translation, rotation, reflection or glide reflection) of another." Here is my Edited. My hand-accounting (now was repeatedly in error, corrected several times) [by Theo, Timothy, Gerhard, and Joel. As suggested by Gerhardobserved, I now have a preliminary computer implementation, which shows that the 18th in my list number of 6-penominoes polyominoes up to$n{=}7\$ heptominoes is a repeat1,1,2,4,10,21,49, but I cannot edit the image today):

I checked a sequence not (yet!) in the OEISfor 1,1,2,4,10,20, but it is unrevealing (to me). Clearly "Sorry, but the terms do not match anything in the table.") I need to count apologize for the polyominoes inadequacy of seven squares to narrow the options. But this is a nontrivialfollowing displays, error-prone undertaking.

Does anyone recognize this enumeration? which I have found countings of various "lattice animals," but not of these particular creatures. Even asymptotics would be useful.

Thanks!will improve eventually:
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4:
10:
21:
49:

7 added 108 characters in body
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2 Corrected as per Theo Buler's remarks.
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