A183986 T(n,k) = 1/4 the number of (n+1) X (k+1) binary arrays with all 2 X 2 subblock sums the same.

4, 6, 6, 9, 8, 9, 15, 11, 11, 15, 25, 17, 14, 17, 25, 45, 27, 20, 20, 27, 45, 81, 47, 30, 26, 30, 47, 81, 153, 83, 50, 36, 36, 50, 83, 153, 289, 155, 86, 56, 46, 56, 86, 155, 289, 561, 291, 158, 92, 66, 66, 92, 158, 291, 561, 1089, 563, 294, 164, 102, 86, 102, 164, 294, 563

Offset

1,1

Comments

Table starts

...4...6...9..15..25..45..81.153.289..561.1089.2145.4225.8385.16641.33153.66049

...6...8..11..17..27..47..83.155.291..563.1091.2147.4227.8387.16643.33155.66051

...9..11..14..20..30..50..86.158.294..566.1094.2150.4230.8390.16646.33158.66054

..15..17..20..26..36..56..92.164.300..572.1100.2156.4236.8396.16652.33164.66060

..25..27..30..36..46..66.102.174.310..582.1110.2166.4246.8406.16662.33174.66070

..45..47..50..56..66..86.122.194.330..602.1130.2186.4266.8426.16682.33194.66090

..81..83..86..92.102.122.158.230.366..638.1166.2222.4302.8462.16718.33230.66126

.153.155.158.164.174.194.230.302.438..710.1238.2294.4374.8534.16790.33302.66198

.289.291.294.300.310.330.366.438.574..846.1374.2430.4510.8670.16926.33438.66334

.561.563.566.572.582.602.638.710.846.1118.1646.2702.4782.8942.17198.33710.66606

Links

R. H. Hardin, Table of n, a(n) for n = 1..683

Formula

Empirical, for every row and column: a(n) = 3*a(n-1)-6*a(n-3)+4*a(n-4).

From Andrew Howroyd, Mar 09 2024: (Start)

The above empirical formula is correct.

T(n,k) = -2 + 2^(n-1) + 2^(k-1) + 2^(floor((n-1)/2)) + 2^(floor(n/2)) + 2^(floor((k-1)/2)) + 2^(floor(k/2)). (End)

Example

Some solutions for 6 X 5
..0..1..0..0..1....1..1..1..1..1....1..1..0..0..1....1..1..0..1..0
..1..0..1..1..0....1..0..1..0..1....0..0..1..1..0....0..0..1..0..1
..0..1..0..0..1....1..1..1..1..1....1..1..0..0..1....1..1..0..1..0
..1..0..1..1..0....0..1..0..1..0....0..0..1..1..0....0..0..1..0..1
..0..1..0..0..1....1..1..1..1..1....1..1..0..0..1....1..1..0..1..0
..1..0..1..1..0....0..1..0..1..0....0..0..1..1..0....0..0..1..0..1

Prog

(PARI) T(n, k) = my(m=2, b=t->2^t-1); m^2 + (m-1)^2*(b(n-1) + b(k-1)) + (m-1)*(b((n-1)\2) + b(n\2) + b((k-1)\2) + b(k\2)) \\ Andrew Howroyd, Mar 09 2024

Crossrefs

Columns 1..8 are A183978, A183979, A183980, A183981, A183982, A183983, A183984, A183985.

Main diagonal is A183977.

Cf. A184039, A184048.

Sequence in context: A201775 A199172 A323674 * A160995 A215454 A155750

Adjacent sequences: A183983 A183984 A183985 * A183987 A183988 A183989

Keyword

nonn,tabl

Author

R. H. Hardin, Jan 08 2011

Status

approved