A335711 The number of free polyominoes of width 2 and height n.

2, 6, 12, 30, 65, 158, 362, 875, 2064, 4984, 11914, 28764, 69155, 166956, 402372, 971413, 2343518, 5657754, 13654968, 32966010, 79577189, 192116330, 463786190, 1119678911, 2703086892, 6525829036, 15754607062, 38034986040, 91824246215, 221683340568, 535190123592

Offset: 2

Author

R. J. Mathar, Jun 18 2020

Keywords

• nonn

Comments

The second column of A268371.

Examples

			a(2)=2, bounding box 2 X 2, counts the L-shaped 3-omino and the full block 4-omino.
a(3)=6, bounding box 2 X 3, counts three 4-ominoes, two 5-omioes, and the full 2 X 3 block 6-omino.
a(4)=12, bounding box 2 X 4, counts three 5-ominoes, six 6-ominoes, two 7-ominoes, and the full 2 X 4 block 8-omino.
		

Links

• John Mason, Table of n, a(n) for n = 2..1000
• Index entries for sequences related to polyominoes

Crossrefs

Cf. A268371, A107769 (asymmetric), A005409 (C_2 symmetry and higher), A352720 (width 2 and size n).

Formula

Conjecture: a(n) = A107769(n-1) + A005409(floor((n+3)/2)).

Conjectures from Colin Barker, Jun 24 2020: (Start)

G.f.: x^2*(2 - 8*x^2 + 2*x^3 - x^4 + x^5 + x^6) / ((1 - x)*(1 - 2*x - x^2)*(1 - 2*x^2 - x^4)).

a(n) = 3*a(n-1) + a(n-2) - 7*a(n-3) + 3*a(n-4) - a(n-5) + a(n-6) + a(n-7) for n>8.

(End)

a(n) = (2*r(n) + 2*m(n) + A078057(n) + 1) / 4, where r(n) = A078057(floor((n-1)/2) - 1)/2, and m(n) = A078057(floor((n+1)/2) - 3)/2. - John Mason, Feb 28 2022

Extensions

a(12)-a(20) from Jean-Luc Manguin, Jun 23 2020

a(21)-a(28) from John Mason, Feb 27 2022

a(29)-a(32) from John Mason, Feb 28 2022