I was logged in successfully. I just couldn't figure out how to edit the sequence entry. I don't know what I was missing. 2012/4/9 Neil Sloane <njasloane@gmail.com>
Allan, I updated A147680.
(But you certainly could have done it yourself. I don't know what the problem could have been. Perhaps take a look at the web page on the OEIS Wiki called
- * Trouble registering, logging in, changing password< https://oeis.org/wiki/Trouble_registering,_logging_in,_changing_password> *
Possibly you didn't log in, which you do at the little login button at the top right of any OEIS page. Your login name is Allan C. Wechsler, perhaps you left out the C? Possible your browser did an automatic fill that did not include the C?)
Best regards
Neil
2012/4/9 Allan Wechsler <acwacw@gmail.com>
A long time ago I proposed a definition for "disk polyominoes", which are essentially rasterized disks. I was able to enumerate these polyominoes (as usual, considering congruent polyominoes to be identical) up to order 11, which was just enough to run off the end of OEIS. Neil was kind enough to key the new sequence in as A147680. Later I was able to add a[12], which Neil added for me. These 13 elements were: 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 3, 3, 4.
Still later, I managed to calculate a[13] = 4, but apparently I didn't inform anybody of this. At this point the clunkiness of my search methods caught up with me, and I didn't have the patience to work out a[14]. Finally, today, I can announce that a[14] = 4. I have some more powerful lemmas under my belt now, and in a few days I ought to be able to come up with a[15] as well.
The actual polyominoes for order 13, in the same notation described earlier in this thread, are (77f7), (6ff7), (27ff2), and (4eve4). I apologize for that "v": it represents a decimal 31, binary 11111, a column of five lattice-points.
The four disk polyominoes of order 14 are (7ff7), (2fff2), (27ff6), and (4eve6).
Now, a meta-question: I have an account at the OEIS wiki, but I wasn't able to figure out how to enter this update for myself. What am I missing?
If there is any interest, I will present my lemmas here, so other people can join the hunt for disk polyominoes. The sequence seems to grow quite slowly, and I have conflicting intuitions about whether the growth rate is polynomial.
2009/5/2 Allan Wechsler <acwacw@gmail.com>
I assume you would then define a[n] as the number of such maximal polyominoes containing n lattice points. For small n, starting at n=0, I get 0, 1, 1, 0, 1, 1. But a[6] has me temporarily stalled. We clearly have to have d=sqrt(5), but so far the only maximal polyomino I have found with that diameter is (373), which has 7 points. I suspect a[6] = 0, a[7] = 1, but I'm not sure yet.
2009/5/2 victor miller <victorsmiller@gmail.com>
A related seqiuence to look at would be the following:
For each d>0 look at sets of lattice points maximal with respect to the property that every 2 of them is distance <=d apart. For these type of polyominoes you don't have to specify a center. It's not clear that is identical with the circular polyominoes since ther are non-circular regions of constant diameter.
Victor
On 5/2/09, Allan Wechsler <acwacw@gmail.com> wrote:
Here's a reference I can't consult right away, which seems at least partly germane:
Srečko Brlek, Gilbert Labelle, and Annie Lacasse; On Minimal Moment of Inertia Polyominoes; in Discrete Geometry For Computer Imagery, pp. 299-309. Springer, Berlin, 2008.
On Sat, May 2, 2009 at 6:24 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Neil was kind enough to enter the sequence for me; the OEIS sequence number is A147680. He asks for some pictures, but I confess that what I've got doesn't actually show circles. Typically I've drawn a bunch of lattice points, highlighting the ones that lie on the circumference of the smallest containing circle. Other marked points need to lie outside that circle in order for the disk polyomino to be valid, but I'm not sure how illuminating my diagrams would be. Perhaps a picture of the lone 7-point example with its bounding circle and other points clearly lying outside would convey the idea, but I don't have a nice one.
I should probably list the polyominoes I've shown to be disks. I could use the notation we used to use for small Life patterns, where each row is represented by the value of a binary number whose ones show which points are part of the configuration. These numbers are usually small, and we write the different row-descriptors with no delimiter between them, going up to letters of the alphabet if we run out of digits. We usually pick a scan order that minimizes the maximum descripton.
For order 0, we of course have only (0), and for order 1 only (1). Order 2 gives (11), and order 3 gives the L-tromino (13). Order 4 has two examples, the block (33) and the T-tetromino (131). Order 5 gives the P-pentomino (133) and the X-pentomino (272).
Order 6: (273), (333).
Order 7: (373).
Order 8: (377), (2772).
Order 9: (777), (2773).
Order 10: (2777), (3773), (27f6). (That "f" means 15, with four adjacent points in a row included in the polyomino.)
Order 11: (3777), (27f7), (67f6).
I only have 80% confidence that these lists are exhaustive. I'm 99% confident that all the polyominoes listed are in fact of the disk type.
I leave you with a puzzle: Is the duodecomino (67f7) a disk polyomino? (I suspect RWG can generate much more fiendish conundra of this variety.)
On Fri, May 1, 2009 at 2:12 PM, Allan Wechsler <acwacw@gmail.com> wrote:
> A funster who might or might not wish to remain anonymous has corrected > me > in private. A3 does not equal 2; it equals 1. The corrected sequence is: > > 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 3, 3 ... > > As before, I'm not entirely confident of the last couple of values. > Thank > you for the correction. >
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