A028247 Number of T-frame polyominoes with n cells.

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

Offset: 1

Author

Anne Fontaine (fonta(AT)hvcc.edu), Hudson Valley Community College, Troy NY 12180

Keywords

• nonn

Comments

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

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, Jan 31 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.

Examples

			The a(6) = 6 polyominoes are:
   OOO   OOO   OOOO   OOOO  OOOOO  OOOOO
    O     OO    O      OO    O       O
    O     O     O
    O
		

Links

• John Mason, Table of n, a(n) for n = 1..1000

Crossrefs

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

Programs

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

Formula

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

Extensions

a(1)-a(3) and terms a(32) and beyond from Allan C. Wechsler and John Mason, Feb 03 2023