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A184039
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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.
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12
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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FORMULA
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Empirical, for all rows and columns: a(n)=3*a(n-1)-6*a(n-3)+4*a(n-4).
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))). - Andrew Howroyd, Mar 09 2024
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EXAMPLE
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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
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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