A360420 -id:A360420 - cp's OEIS frontend

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

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

nonn

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.

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

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

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

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

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

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

A360419 a(n) = the number of U-frame polyominoes with n cells, reduced for symmetry.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 5, 9, 16, 24, 37, 50, 71, 93, 121, 151, 192, 231, 285, 338, 398, 470, 548, 626, 723, 827, 924, 1056, 1175, 1314, 1454, 1629, 1763, 1985, 2138, 2356, 2540, 2820, 2976, 3305, 3491, 3834, 4039, 4441, 4613, 5103, 5291, 5775, 5999, 6572

Offset: 1

John Mason, Feb 06 2023

nonn

A U-frame polyomino has a perimeter that forms a self-avoiding polygon such that as you traverse the perimeter counterclockwise you encounter turns in the order LLLLLLRR.

			a(5)=1 because of:
  OO
  O
  OO
The a(7) = 5 polyominoes are:
  O
  O     O O             O
  O O   O O   O OO   O  O   O   O
  OOO   OOO   OOOO   OOOO   OOOOO

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A028247, A270060, A360420, A360421.

B(n,k,x) = sum(j=k, n, x^j/(1 - x^j), O(x*x^n))
seq(n) = Vec(sum(k=1, (n-2)\3, x^k*(B(n-k, k+1, x)^2 + B((n-k)\2, k+1, x^2))/(1-x^k), O(x*x^n))/2, -n) \\ Andrew Howroyd, Feb 07 2023

G.f.: Sum_{k>=1} (x^k/(1 - x^k)) * (B(k+1, x)^2 + B(k+1, x^2))/2 where B(k, x) = Sum_{j>=k} x^j/(1 - x^j). - Andrew Howroyd, Feb 07 2023

A360421 a(n) = the number of X-frame polyominoes with n cells, reduced for symmetry.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 7, 20, 49, 109, 216, 414, 723, 1228, 1958, 3065, 4580, 6734

Offset: 1

John Mason, Feb 06 2023

An X-frame polyomino has a perimeter that forms a self-avoiding polygon such that as you traverse the perimeter counterclockwise you encounter turns in the order LLRLLRLLRLLR.

			a(5) = 1 because of:
    O
   OOO
    O
The a(7) = 7 polyominoes are:
    O      O      O       O     O     O     O
   OOOOO  OOOOO  OOOOO  OOOOO  OOOO  OOOO  OOOO
    O       O       O     O     O      O    OO
                                O      O

Cf. A028247, A270060, A360419, A360420.

cp's OEIS Frontend

A028247 Number of T-frame polyominoes with n cells.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A360419 a(n) = the number of U-frame polyominoes with n cells, reduced for symmetry.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A360421 a(n) = the number of X-frame polyominoes with n cells, reduced for symmetry.

Views

Author

Keywords

Comments

Examples

Crossrefs