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The number of free polyominoes of width 2 and height n.
4

%I #41 Nov 09 2022 19:17:13

%S 2,6,12,30,65,158,362,875,2064,4984,11914,28764,69155,166956,402372,

%T 971413,2343518,5657754,13654968,32966010,79577189,192116330,

%U 463786190,1119678911,2703086892,6525829036,15754607062,38034986040,91824246215,221683340568,535190123592

%N The number of free polyominoes of width 2 and height n.

%C The second column of A268371.

%H John Mason, <a href="/A335711/b335711.txt">Table of n, a(n) for n = 2..1000</a>

%H <a href="/index/Pol#polyominoes">Index entries for sequences related to polyominoes</a>

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

%F Conjectures from _Colin Barker_, Jun 24 2020: (Start)

%F 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)).

%F 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.

%F (End)

%F 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

%e a(2)=2, bounding box 2 X 2, counts the L-shaped 3-omino and the full block 4-omino.

%e a(3)=6, bounding box 2 X 3, counts three 4-ominoes, two 5-omioes, and the full 2 X 3 block 6-omino.

%e 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.

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

%K nonn

%O 2,1

%A _R. J. Mathar_, Jun 18 2020

%E a(12)-a(20) from _Jean-Luc Manguin_, Jun 23 2020

%E a(21)-a(28) from _John Mason_, Feb 27 2022

%E a(29)-a(32) from _John Mason_, Feb 28 2022