A184039 T(n,k) = 1/16 the number of (n+1) X (k+1) 0..3 arrays with all 2 X 2 subblocks having the same four values.

16, 28, 28, 49, 40, 49, 91, 61, 61, 91, 169, 103, 82, 103, 169, 325, 181, 124, 124, 181, 325, 625, 337, 202, 166, 202, 337, 625, 1225, 637, 358, 244, 244, 358, 637, 1225, 2401, 1237, 658, 400, 322, 400, 658, 1237, 2401, 4753, 2413, 1258, 700, 478, 478, 700, 1258

Offset

1,1

Comments

Table starts

...16...28...49...91..169..325..625.1225.2401.4753..9409.18721.37249.74305

...28...40...61..103..181..337..637.1237.2413.4765..9421.18733.37261.74317

...49...61...82..124..202..358..658.1258.2434.4786..9442.18754.37282.74338

...91..103..124..166..244..400..700.1300.2476.4828..9484.18796.37324.74380

..169..181..202..244..322..478..778.1378.2554.4906..9562.18874.37402.74458

..325..337..358..400..478..634..934.1534.2710.5062..9718.19030.37558.74614

..625..637..658..700..778..934.1234.1834.3010.5362.10018.19330.37858.74914

.1225.1237.1258.1300.1378.1534.1834.2434.3610.5962.10618.19930.38458.75514

.2401.2413.2434.2476.2554.2710.3010.3610.4786.7138.11794.21106.39634.76690

.4753.4765.4786.4828.4906.5062.5362.5962.7138.9490.14146.23458.41986.79042

Links

R. H. Hardin, Table of n, a(n) for n = 1..287

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) = -14 + 9*(2^(n-1) + 2^(k-1)) + 3*(2^(floor((n-1)/2)) + 2^(floor(n/2)) + 2^(floor((k-1)/2)) + 2^(floor(k/2))). (End)

Example

Some solutions for 4X3
..0..3..0....3..2..3....3..2..3....1..0..1....2..3..2....3..2..2....3..1..3
..3..2..3....3..3..3....1..3..1....2..1..2....3..2..3....2..1..3....1..2..1
..3..0..3....3..2..3....3..2..3....0..1..0....2..3..2....3..2..2....3..1..3
..2..3..2....3..3..3....1..3..1....1..2..1....2..3..2....2..1..3....2..1..2

Prog

(PARI) T(n, k) = my(m=4, 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 A184031, A184032, A184033, A184034, A184035, A184036, A184037, A184038.

Main diagonal is A184030.

Cf. A183986, A184048.

Sequence in context: A300132 A101857 A350302 * A212051 A166595 A280989

Adjacent sequences: A184036 A184037 A184038 * A184040 A184041 A184042

Keyword

nonn,tabl

Author

R. H. Hardin, Jan 08 2011

Status

approved