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reviewed

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Equivalently, polyominoes which are integral rectangles with integral notches cut from two adjacent corners; or right-angled octagons with integral sides, and as you traverse the perimeter counterclockwise you encounter turns in the order LLLLRLLR. - Allan C. Wechsler, from seqfans mailing list Feb, Jan 0631 2023.

For 2 <= n <= 28, a(2n) < a(2n+1); for 29 <= n <= 99, a(2n) > a(2n+1). - Don Reble from seqfans email, Jan 31 2023.

For 2 <= n <= 28, a(2n) < a(2n+1); for 29 <= n <= 99, a(2n) > a(2n+1). - Don Reble in seqfans email, Jan 31 2023

John Mason: Completed with the union of David's and my edits. I lost the java program as superseded by Andrew's posts.

AlternativelyEquivalently, polyominoes which are integral rectangles with integral notches cut from two adjacent corners; or right-angled octagons itwith isintegral asides, and freeas convexyou polyominotraverse whosethe perimeter counterclockwise you hasencounter aturns braceletin equalthe toorder BBWBBBBWLLLLRLLR. - _JohnAllan MasonC. Wechsler_, from seqfans mailing list Feb 0306 2023.

John Mason, <a href="/A028247/a028247_1.txt">Java program</a>

proposed

editing

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Michel Marcus: AUTHOR should simply be : Anne Fontaine (fonta(AT)hvcc.edu)  ??

N. J. A. Sloane: No, leave her name intact - especially since we have been unable to read her.  To shorten the author line as you propose would be a sin.  It would lose valuable information.  Surely that is obvious. The most important rule for an editor is,  first, do no harm.

Michel Marcus: ok Neil as you wish

John Mason: After David's and my edits got mixed up, I suggested to David that he propose his own changes (T-frame definition) and complete the approval process, and then I would introduce my edits (b-file, program, xrefs) afterwards. To tell the truth, now that Andrew has put in a formula and Pari, I would quite happily see my java program disappear. The question remains - who should take over the responsibility for editing now?

Andrew Howroyd: John, I think it is really up to you. Either edit now, or wait until approved and then re-edit. (I am not going to touch anything more now, so there is no risk of you stepping on my toes). Unfortunately, I didn't see your original edit until after I had already done mine - hence triple conflation of edits.

A T-frame is a polyomino whose boundary word has the form x^a y^b x^c y^d x^-e y^-f x^g y^-h, where a, b, c, d, e, f, g, h are positive integers. The boundary word is determined by moving counterclockwise around the boundary of the polyomino. The symbols x and y represent unit steps to the right and up, respectively, while x^-1 and y^-1 represent steps to the left and down.. - _David Radcliffe_, Jan 31 2023

- David Radcliffe, from seqfans mailing list, 2023.

Equivalently, polyominoes which are integral rectangles with integral notches cut from two adjacent corners; or right-angled octagons with integral sides, and as you traverse the perimeter counterclockwise you encounter turns in the order LLLLRLLR.

- Allan Wechsler, from seqfans mailing list, 2023.

Alternatively it is a free convex polyomino whose perimeter has a bracelet equal to BBWBBBBW. - John Mason, Feb 03 2023

AndrewJohn HowroydMason, <a href="/A028247/b028247_1.txt">Table of n, a(n) for n = 1..1000</a>

John Mason, <a href="/A028247/a028247_1.txt">Java program</a>

For 2 <= n <= 28, a(2n) < a(2n+1); for 29 <= n <= 99, a(2n) > a(2n+1). - _Don Reble__ in seqfans email, Jan 31 2023

a(1)-a(3) and terms a(32) and beyond from _Allan C. Wechsler_ and _AndrewJohn HowroydMason_, Feb 0803 2023

Andrew Howroyd: John and Allen loaded the b-file and even a program. I suggest we aim to review / approve this sooner rather than later. (& then anyone dissatisfied can start a new edit)

G.f.: Sum_{k>=2} (x^k/(1-x^k)) * (B(k-1, x)^2 + B(k-1, x^2))/2 where B(k,x) = Sum_{j=1..k} x^j/(1-x^j). - Andrew Howroyd, Feb 08 2023

(PARI) B(k, x) = sum(j=1, k, x^j/(1-x^j))

seq(n) = Vec(sum(k=2, n, (x^k/(1-x^k)) * (B(k-1, x + O(x^(1+n-k)))^2 + B(k-1, x^2 + O(x^(1+n-k))))/2, O(x*x^n)), -n) \\ Andrew Howroyd, Feb 08 2023

Andrew Howroyd: I don't really know what happened here. David has reverted a significant edit by others. (See history). I will attempt repair.

0, 0, 0, 1, 2, 6, 10, 19, 28, 44, 60, 86, 110, 146, 182, 233, 278, 343, 403, 490, 557, 664, 749, 879, 978, 1132, 1237, 1435, 1551, 1771, 1905, 2168, 2296, 2608, 2758, 3101, 3256, 3655, 3798, 4274, 4419, 4936, 5087, 5670, 5809, 6472, 6602, 7339, 7462, 8271

Andrew Howroyd, <a href="/A028247/b028247_1.txt">Table of n, a(n) for n = 1..1000</a>

a(1)-a(3)=0) and terms a(32) and prependedbeyond byfrom Andrew Howroyd, Feb 08 2023

0, 0, 0, 1, 2, 6, 10, 19, 28, 44, 60, 86, 110, 146, 182, 233, 278, 343, 403, 490, 557, 664, 749, 879, 978, 1132, 1237, 1435, 1551, 1771, 1905

4,2

1,5

The a(6) = 6 polyominoes are:

OOO OOO OOOO OOOO OOOOO OOOOO

O OO O OO O O

O O O

O

Cf. A270060 (L frame), A360419 (U frame), A360420 (Z frame).

a(1)-a(3)=0 prepended by Andrew Howroyd, Feb 08 2023

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editing