|
%I #11 Mar 09 2024 16:25:45
%S 16,28,28,49,40,49,91,61,61,91,169,103,82,103,169,325,181,124,124,181,
%T 325,625,337,202,166,202,337,625,1225,637,358,244,244,358,637,1225,
%U 2401,1237,658,400,322,400,658,1237,2401,4753,2413,1258,700,478,478,700,1258
%N 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.
%C Table starts
%C ...16...28...49...91..169..325..625.1225.2401.4753..9409.18721.37249.74305
%C ...28...40...61..103..181..337..637.1237.2413.4765..9421.18733.37261.74317
%C ...49...61...82..124..202..358..658.1258.2434.4786..9442.18754.37282.74338
%C ...91..103..124..166..244..400..700.1300.2476.4828..9484.18796.37324.74380
%C ..169..181..202..244..322..478..778.1378.2554.4906..9562.18874.37402.74458
%C ..325..337..358..400..478..634..934.1534.2710.5062..9718.19030.37558.74614
%C ..625..637..658..700..778..934.1234.1834.3010.5362.10018.19330.37858.74914
%C .1225.1237.1258.1300.1378.1534.1834.2434.3610.5962.10618.19930.38458.75514
%C .2401.2413.2434.2476.2554.2710.3010.3610.4786.7138.11794.21106.39634.76690
%C .4753.4765.4786.4828.4906.5062.5362.5962.7138.9490.14146.23458.41986.79042
%H R. H. Hardin, <a href="/A184039/b184039.txt">Table of n, a(n) for n = 1..287</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) = -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))). - _Andrew Howroyd_, Mar 09 2024
%e Some solutions for 4X3
%e ..0..3..0....3..2..3....3..2..3....1..0..1....2..3..2....3..2..2....3..1..3
%e ..3..2..3....3..3..3....1..3..1....2..1..2....3..2..3....2..1..3....1..2..1
%e ..3..0..3....3..2..3....3..2..3....0..1..0....2..3..2....3..2..2....3..1..3
%e ..2..3..2....3..3..3....1..3..1....1..2..1....2..3..2....2..1..3....2..1..2
%o (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
%Y Columns 1..8 are A184031, A184032, A184033, A184034, A184035, A184036, A184037, A184038.
%Y Main diagonal is A184030.
%Y Cf. A183986, A184048.
%K nonn,tabl
%O 1,1
%A _R. H. Hardin_, Jan 08 2011
|
