A184048 T(n,k) = 1/9 the number of (n+1) X (k+1) 0..2 arrays with all 2 X 2 subblocks having the same four values.

9, 15, 15, 25, 21, 25, 45, 31, 31, 45, 81, 51, 41, 51, 81, 153, 87, 61, 61, 87, 153, 289, 159, 97, 81, 97, 159, 289, 561, 295, 169, 117, 117, 169, 295, 561, 1089, 567, 305, 189, 153, 189, 305, 567, 1089, 2145, 1095, 577, 325, 225, 225, 325, 577, 1095, 2145, 4225, 2151

Offset

1,1

Comments

Table starts

....9...15...25...45...81..153..289..561.1089.2145.4225..8385.16641.33153.66049

...15...21...31...51...87..159..295..567.1095.2151.4231..8391.16647.33159.66055

...25...31...41...61...97..169..305..577.1105.2161.4241..8401.16657.33169.66065

...45...51...61...81..117..189..325..597.1125.2181.4261..8421.16677.33189.66085

...81...87...97..117..153..225..361..633.1161.2217.4297..8457.16713.33225.66121

..153..159..169..189..225..297..433..705.1233.2289.4369..8529.16785.33297.66193

..289..295..305..325..361..433..569..841.1369.2425.4505..8665.16921.33433.66329

..561..567..577..597..633..705..841.1113.1641.2697.4777..8937.17193.33705.66601

.1089.1095.1105.1125.1161.1233.1369.1641.2169.3225.5305..9465.17721.34233.67129

.2145.2151.2161.2181.2217.2289.2425.2697.3225.4281.6361.10521.18777.35289.68185

Links

R. H. Hardin, Table of n, a(n) for n = 1..391

Formula

Empirical, for all rows and columns: 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) = -7 + 4*(2^(n-1) + 2^(k-1)) + 2*(2^(floor((n-1)/2)) + 2^(floor(n/2)) + 2^(floor((k-1)/2)) + 2^(floor(k/2))). (End)

Example

Some solutions for 6X5
..0..1..0..1..0....1..2..1..2..1....1..2..1..2..1....2..1..2..2..2
..0..1..0..1..0....1..2..1..2..1....1..0..1..0..1....0..2..0..1..0
..1..0..1..0..1....2..1..2..1..2....1..2..1..2..1....2..1..2..2..2
..0..1..0..1..0....1..2..1..2..1....1..0..1..0..1....0..2..0..1..0
..0..1..0..1..0....1..2..1..2..1....2..1..2..1..2....2..1..2..2..2
..1..0..1..0..1....2..1..2..1..2....0..1..0..1..0....0..2..0..1..0

Prog

(PARI) T(n, k) = my(m=3, 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(n+2), A184041, A184042, A184043, A184044, A184045, A184046, A184047.

Main diagonal is A184040.

Cf. A183986, A184039.

Sequence in context: A130119 A346609 A232395 * A284128 A058957 A257409

Adjacent sequences: A184045 A184046 A184047 * A184049 A184050 A184051

Keyword

nonn,tabl

Author

R. H. Hardin, Jan 08 2011

Status

approved