|
%I #29 Feb 12 2023 06:30:37
%S 0,0,0,1,2,6,10,19,28,44,60,86,110,146,182,233,278,343,403,490,557,
%T 664,749,879,978,1132,1237,1435,1551,1771,1905,2168,2296,2608,2758,
%U 3101,3256,3655,3798,4274,4419,4936,5087,5670,5809,6472,6602,7339,7462,8271
%N Number of T-frame polyominoes with n cells.
%C 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
%C 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.
%C 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.
%H John Mason, <a href="/A028247/b028247.txt">Table of n, a(n) for n = 1..1000</a>
%F 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
%e The a(6) = 6 polyominoes are:
%e OOO OOO OOOO OOOO OOOOO OOOOO
%e O OO O OO O O
%e O O O
%e O
%o (PARI) B(k,x) = sum(j=1, k, x^j/(1-x^j))
%o 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
%Y Cf. A270060 (L frame), A360419 (U frame), A360420 (Z frame).
%K nonn
%O 1,5
%A Anne Fontaine (fonta(AT)hvcc.edu), Hudson Valley Community College, Troy NY 12180.
%E a(1)-a(3) and terms a(32) and beyond from _Allan C. Wechsler_ and _John Mason_, Feb 03 2023
|
