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A335711
The number of free polyominoes of width 2 and height n.
4
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
2,1
COMMENTS
The second column of A268371.
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
EXAMPLE
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.
CROSSREFS
Cf. A268371, A107769 (asymmetric), A005409 (C_2 symmetry and higher), A352720 (width 2 and size n).
Sequence in context: A283477 A182863 A161507 * A032177 A095349 A291518
KEYWORD
nonn
AUTHOR
R. J. Mathar, Jun 18 2020
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
STATUS
approved