A183304 -id:A183304 - cp's OEIS frontend

A183312 T(n,k) = Half the number of n X k binary arrays with no element equal to a strict majority of its horizontal and vertical neighbors.

1, 1, 1, 1, 3, 1, 2, 4, 4, 2, 3, 6, 9, 6, 3, 5, 9, 19, 19, 9, 5, 8, 14, 42, 55, 42, 14, 8, 13, 22, 93, 178, 178, 93, 22, 13, 21, 35, 205, 572, 910, 572, 205, 35, 21, 34, 56, 452, 1798, 4212, 4212, 1798, 452, 56, 34, 55, 90, 997, 5700, 19899, 29400, 19899, 5700, 997, 90, 55, 89

Offset: 1

R. H. Hardin, Jan 03 2011

Same solutions for no element unequal to a strict majority of its horizontal and vertical neighbors, via xor with a 0101... checkerboard pattern.
Table starts
..1..1....1.....2.......3........5..........8..........13............21
..1..3....4.....6.......9.......14.........22..........35............56
..1..4....9....19......42.......93........205.........452...........997
..2..6...19....55.....178......572.......1798........5700.........18064
..3..9...42...178.....910.....4212......19899.......94217........445859
..5.14...93...572....4212....29400.....206755.....1447110......10149621
..8.22..205..1798...19899...206755....2160250....22504107.....234636215
.13.35..452..5700...94217..1447110...22504107...348871589....5406312318
.21.56..997.18064..445859.10149621..234636215..5406312318..124597748299
.34.90.2199.57249.2113257.71244598.2447317278.83828453334.2872365166632

			Some solutions with a(1,1)=0 for 6 X 6
..0..1..0..1..0..0....0..1..1..0..0..1....0..1..1..0..0..1....0..1..0..1..0..0
..1..1..0..0..1..1....1..0..0..1..1..0....1..0..0..1..1..0....0..1..1..0..1..1
..0..0..1..1..0..0....0..1..0..0..1..0....0..0..1..0..0..1....1..0..1..1..0..0
..1..1..1..0..1..1....0..1..1..0..0..1....1..1..0..1..1..0....0..1..0..0..1..1
..0..0..0..1..0..0....1..0..0..1..0..0....0..1..0..0..1..0....1..0..0..1..0..0
..1..1..0..1..0..1....0..1..1..0..1..1....1..0..1..1..0..1....1..0..1..0..1..1

R. H. Hardin, Table of n, a(n) for n = 1..337

Column 1 is A000045(n-1) for n>1.
Column 2 is A000045(n+1)+1 for n>1.
Cf. A183304 (col 3), A183305 (col 4), A183306 (col 5), A183307 (col 6), A183308 (col 7), A183309 (col 8).

A084481 Number of fault-free tilings of a 4 X 2n rectangle with L tetrominoes.

Original entry on oeis.org

2, 6, 10, 18, 38, 84, 186, 410, 904, 1994, 4398, 9700, 21394, 47186, 104072, 229538, 506262, 1116596, 2462730, 5431722, 11980040, 26422810, 58277342, 128534724, 283492258, 625261858, 1379058440, 3041609138, 6708480134, 14796018708, 32633646554, 71975773242

Offset: 1

Ralf Stephan, May 27 2003

Fault-free tilings are those where the only straight interface is at the left and right end. Thus a(n) <= A084480(n).

If the conjectured G.F. in A183304 is true, then a(n)= 2*A183304(n-1), n>3. - R. J. Mathar, Dec 02 2022

Colin Barker, Table of n, a(n) for n = 1..1000
Nicolas Bělohoubek and Antonín Slavík, L-Tetromino Tilings and Two-Color Integer Compositions, Univ. Karlova (Czechia, 2025). See p. 10.
C. Moore, [math/9905012] Some Polyomino Tilings of the Plane
Index entries for linear recurrences with constant coefficients, signature (2,0,1).

Cf. A084478, A084479, A084480, A084477.

Vec(2*x*(1 + x)^2*(1 - x - x^3) / (1 - 2*x - x^3) + O(x^30)) \\ Colin Barker, Mar 28 2017

G.f.: 2*z*(1+z)^2*(1-z-z^3) / (1-2*z-z^3).

a(n) = 2*a(n-1) + a(n-3) for n>6. - Colin Barker, Mar 28 2017

cp's OEIS Frontend

A183312 T(n,k) = Half the number of n X k binary arrays with no element equal to a strict majority of its horizontal and vertical neighbors.

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