A184048 - OEIS

%I #8 Mar 09 2024 16:25:40

%S 9,15,15,25,21,25,45,31,31,45,81,51,41,51,81,153,87,61,61,87,153,289,

%T 159,97,81,97,159,289,561,295,169,117,117,169,295,561,1089,567,305,

%U 189,153,189,305,567,1089,2145,1095,577,325,225,225,325,577,1095,2145,4225,2151

%N 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.

%C Table starts

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

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

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

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

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

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

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

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

%C .1089.1095.1105.1125.1161.1233.1369.1641.2169.3225.5305..9465.17721.34233.67129

%C .2145.2151.2161.2181.2217.2289.2425.2697.3225.4281.6361.10521.18777.35289.68185

%H R. H. Hardin, <a href="/A184048/b184048.txt">Table of n, a(n) for n = 1..391</a>

%F Empirical, for all rows and columns: a(n)=3*a(n-1)-6*a(n-3)+4*a(n-4).

%F From _Andrew Howroyd_, Mar 09 2024: (Start)

%F The above empirical formula is correct.

%F 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)

%e Some solutions for 6X5

%e ..0..1..0..1..0....1..2..1..2..1....1..2..1..2..1....2..1..2..2..2

%e ..0..1..0..1..0....1..2..1..2..1....1..0..1..0..1....0..2..0..1..0

%e ..1..0..1..0..1....2..1..2..1..2....1..2..1..2..1....2..1..2..2..2

%e ..0..1..0..1..0....1..2..1..2..1....1..0..1..0..1....0..2..0..1..0

%e ..0..1..0..1..0....1..2..1..2..1....2..1..2..1..2....2..1..2..2..2

%e ..1..0..1..0..1....2..1..2..1..2....0..1..0..1..0....0..2..0..1..0

%o (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

%Y Columns 1..8 are A183978(n+2), A184041, A184042, A184043, A184044, A184045, A184046, A184047.

%Y Main diagonal is A184040.

%Y Cf. A183986, A184039.

%K nonn,tabl

%O 1,1

%A _R. H. Hardin_, Jan 08 2011