[seqfan] Re: Sequence proposal by John Mason (by way of moderator)

[M. F. Hasler]

I like very much Frank's prescription for coding 2-dimensional shapes,
here polyominoes,
by adding those powers of 2, filling the quarter-plane in the
"(OEIS-)canonical" way (antidiagonals), which are "covered" by the
shape (in the way which yields the smallest result).
For the polyominoes this would yield
1 (.) ; 3 (..) ; 7 (:.), 11(...) ; 15 (:..), 23 (::), 27 (.:.), 30
(.:*), 75 (....) ; &c
The sequence
1,3,7,11,15,23,27,30,75, ...
is not yet in OEIS, I can submit this (with due credits to Frank)
unless s/o already started the work.

(Thereafter, the sequence will no more be increasing,
if one follows the prescription to list triominos, 4-ominos, 5-ominos,...:
The first element in each series (or row, if considered as table)
would be 2^n-1, and the last element
woud be 2^0+2^1+2^3+2^6+...+2^T(n).)
OTOH one could also define and list these "polyomino numbers" which
are those numbers which represent some polyomino (using the above
prescription!(*)),
i.e., the former sequence re-ordered by the size of the terms.)
(*) there would also be another [actually 2 other] sequence[s]
(supersequence[s] of the former), of numbers which represent
polyominos without the restriction:
(a) of removing equivalent polyominos (e.g. there would be " .. "  AND " : ")
(b) of translating them as to touch the x and y axis
(i.e., e.g., the ".." and " : " could lie anywhere in the quarter-plane).

PS: another way of assigning the weigths to the grid points would be
to number them not following antidiagonals, but "filling squares":
0 1 4
3 2 5
8 7 6
etc.
This would lead to a different variants for each of the 4 above sequences.

Maximilian


On Sun, Aug 24, 2014 at 1:43 PM, Frank Adams-Watters
<franktaw at netscape.net> wrote:
> Sorry, I should have said the domino is represented by 3, not 2.
>
>
> Franklin T. Adams-Watters
>
> -----Original Message-----
> From: Frank Adams-Watters <franktaw at netscape.net>
> To: seqfan <seqfan at list.seqfan.eu>
> Sent: Sun, Aug 24, 2014 12:40 pm
> Subject: [seqfan] Re: Sequence proposal by John Mason (by way of moderator)
>
> ... So the monomio would
> be represented by 1, the domino by 2, and the trionimos by 7 and 11. ...
>
>
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>
> Seqfan Mailing list - http://list.seqfan.eu/



-- 
Maximilian