| Displaying 1-8 of 8 results found.
page
1
Number of (n+1) X 3 0..2 arrays with every 2 X 3 or 3 X 2 subblock having exactly three counterclockwise and three clockwise edge increases.
+10
2
60, 246, 1122, 5118, 23346, 106494, 485778, 2215902, 10107954, 46107966, 210323922, 959403678, 4376370546, 19963045374, 91062485778, 415386338142, 1894806719154, 8643260919486, 39426691159122, 179846933956638
FORMULA
Empirical: a(n) = 5*a(n-1) - 2*a(n-2) for n > 3.
Empirical g.f.: 6*x*(2 - x)*(5 - 2*x) / (1 - 5*x + 2*x^2). - Colin Barker, Mar 04 2018
EXAMPLE
Some solutions for n=4:
..0..1..0....0..1..0....1..2..1....1..2..0....0..1..2....2..1..0....2..0..2
..1..2..1....2..0..1....0..1..2....2..1..2....1..0..1....0..2..1....1..2..0
..2..0..2....0..1..0....1..0..1....0..2..0....2..1..2....1..0..2....2..0..1
..0..2..1....2..0..1....0..1..2....1..0..2....0..2..0....2..1..0....0..2..0
..1..0..2....0..2..0....1..0..1....0..1..0....2..0..1....0..2..1....2..0..1
Number of (n+1) X (n+1) 0..2 arrays with every 2 X 3 or 3 X 2 subblock having exactly three counterclockwise and three clockwise edge increases.
+10
1
81, 246, 7812, 580986, 101596896, 41869995708, 40724629633188, 93574975249028022, 508279521493590763140, 6529777647254616589112172, 198475392061658571459051861720, 14277440032279343277552357109481028
EXAMPLE
Some solutions for n=4:
..2..1..0..2..1....0..2..1..2..0....1..2..0..2..0....0..1..0..1..2
..0..2..1..0..2....1..0..2..1..2....0..1..2..1..2....1..0..1..0..1
..1..0..2..1..0....0..2..1..0..1....1..2..0..2..0....0..1..0..1..2
..2..1..0..2..1....1..0..2..1..0....0..1..2..0..2....1..2..1..2..1
..0..2..1..0..2....2..1..0..2..1....1..0..1..2..1....0..1..0..1..2
Number of (n+1) X 2 0..2 arrays with every 2 X 3 or 3 X 2 subblock having exactly three counterclockwise and three clockwise edge increases.
+10
1
81, 60, 162, 486, 1458, 4374, 13122, 39366, 118098, 354294, 1062882, 3188646, 9565938, 28697814, 86093442, 258280326, 774840978, 2324522934, 6973568802, 20920706406, 62762119218, 188286357654, 564859072962, 1694577218886
FORMULA
Empirical: a(n) = 3*a(n-1) for n > 3.
EXAMPLE
Some solutions for n=4:
..1..2....0..2....2..1....2..1....2..0....1..2....2..1....1..2....0..2....1..2
..0..1....1..0....0..2....0..2....0..1....2..0....0..2....2..1....1..0....2..0
..1..0....2..1....2..0....1..0....2..0....0..1....1..0....0..2....2..1....1..2
..0..2....0..2....1..2....2..1....0..2....2..0....2..1....1..0....1..0....0..1
..2..0....2..1....2..0....1..2....1..0....1..2....1..0....2..1....0..1....1..2
Number of (n+1) X 4 0..2 arrays with every 2 X 3 or 3 X 2 subblock having exactly three counterclockwise and three clockwise edge increases.
+10
1
162, 1122, 7812, 54450, 379602, 2646540, 18451530, 128643282, 896895828, 6253122402, 43596523890, 303953253948, 2119150161594, 14774631788466, 103008153195972, 718168802906322, 5007044718948114, 34908919346114028
FORMULA
Empirical: a(n) = 9*a(n-1) - 15*a(n-2) + 6*a(n-3).
Empirical g.f.: 6*x*(27 - 56*x + 24*x^2) / (1 - 9*x + 15*x^2 - 6*x^3). - Colin Barker, Jun 13 2018
EXAMPLE
Some solutions for n=4:
2 1 2 1 2 0 2 0 0 1 2 0 0 1 2 1 0 2 0 2
1 2 0 2 0 1 0 1 1 0 1 2 1 0 1 2 1 0 1 0
2 0 1 0 1 2 1 2 0 2 0 1 2 1 2 0 2 1 2 1
1 2 0 1 0 1 0 1 1 0 1 0 0 2 0 1 0 2 0 2
2 0 2 0 2 0 1 0 2 1 2 1 2 0 1 2 1 0 1 0
Number of (n+1) X 5 0..2 arrays with every 2 X 3 or 3 X 2 subblock having exactly three counterclockwise and three clockwise edge increases.
+10
1
486, 5118, 54450, 580986, 6204438, 66274542, 707982258, 7563227466, 80796885414, 863143206558, 9220857727842, 98505357616986, 1052321399885238, 11241828565189710, 120095162925973650, 1282962828563896554
FORMULA
Empirical: a(n) = 16*a(n-1) - 65*a(n-2) + 92*a(n-3) - 48*a(n-4) + 8*a(n-5).
Empirical g.f.: 6*x*(81 - 443*x + 692*x^2 - 376*x^3 + 64*x^4) / (1 - 16*x + 65*x^2 - 92*x^3 + 48*x^4 - 8*x^5). - Colin Barker, Jun 13 2018
EXAMPLE
Some solutions for n=4:
..2..0..1..2..1....0..1..0..1..2....1..0..1..2..0....0..2..1..2..0
..0..2..0..1..2....1..0..1..0..1....2..1..2..0..2....1..0..2..0..1
..1..0..1..0..1....0..1..0..1..2....0..2..0..2..0....2..1..0..1..0
..2..1..2..1..2....1..2..1..2..1....1..0..1..0..1....1..2..1..0..1
..0..2..1..2..0....0..1..0..1..2....2..1..2..1..0....2..0..2..1..0
Number of (n+1) X 6 0..2 arrays with every 2 X 3 or 3 X 2 subblock having exactly three counterclockwise and three clockwise edge increases.
+10
1
1458, 23346, 379602, 6204438, 101596896, 1664748270, 27284864220, 447232269654, 7330916855718, 120167900345196, 1969792468989480, 32288891319485190, 529280670290557896, 8675989562774602188, 142217172462541690296
FORMULA
Empirical: a(n) = 30*a(n-1) - 291*a(n-2) + 1278*a(n-3) - 2901*a(n-4) + 3519*a(n-5) - 2152*a(n-6) + 516*a(n-7).
EXAMPLE
Some solutions for n=4:
..0..1..2..0..1..2....2..0..1..2..0..2....2..1..0..1..2..0....1..0..1..2..1..2
..1..0..1..2..0..1....0..2..0..1..2..0....0..2..1..2..1..2....2..1..2..0..2..1
..0..1..2..1..2..0....1..0..1..0..1..2....1..0..2..0..2..1....1..2..0..2..1..0
..2..0..1..0..1..2....2..1..2..1..2..0....2..1..0..1..0..2....0..1..2..1..0..1
..0..1..0..1..2..1....0..2..0..2..0..2....1..2..1..0..1..0....1..2..1..2..1..0
Number of (n+1) X 7 0..2 arrays with every 2 X 3 or 3 X 2 subblock having exactly three counterclockwise and three clockwise edge increases.
+10
1
4374, 106494, 2646540, 66274542, 1664748270, 41869995708, 1053631126386, 26519876081106, 667567685148144, 16804916171955954, 423043134930340686, 10649666547770137380, 268094963505457212522, 6749037530485227993642
FORMULA
Empirical: a(n) = 55*a(n-1) -1109*a(n-2) +11330*a(n-3) -67206*a(n-4) +247404*a(n-5) -582440*a(n-6) +881876*a(n-7) -846764*a(n-8) +499200*a(n-9) -172400*a(n-10) +33152*a(n-11) -3264*a(n-12) +128*a(n-13).
EXAMPLE
Some solutions for n=4:
..1..2..1..2..0..1..2....2..0..1..2..0..1..2....1..0..1..2..1..2..1
..0..1..0..1..2..0..1....1..2..0..1..2..0..1....2..1..2..0..2..1..0
..2..0..2..0..1..2..0....2..0..1..2..1..2..0....0..2..0..2..0..2..1
..1..2..0..2..0..1..2....0..2..0..1..2..0..2....1..0..1..0..1..0..2
..2..0..2..1..2..0..1....1..0..2..0..1..2..1....0..1..2..1..0..1..0
Number of (n+1) X 8 0..2 arrays with every 2 X 3 or 3 X 2 subblock having exactly three counterclockwise and three clockwise edge increases.
+10
1
13122, 485778, 18451530, 707982258, 27284864220, 1053631126386, 40724629633188, 1574756199257154, 60905630834498448, 2355823979126396484, 91127113576629203340, 3525019283291091200178, 136357697258861377287582
FORMULA
Empirical: a(n) = 105*a(n-1) -4473*a(n-2) +105654*a(n-3) -1580037*a(n-4) +16082595*a(n-5) -116408090*a(n-6) +616031475*a(n-7) -2425718097*a(n-8) +7181467374*a(n-9) -16063429749*a(n-10) +27150592284*a(n-11) -34525830041*a(n-12) +32730104655*a(n-13) -22785383754*a(n-14) +11380763892*a(n-15) -3932260752*a(n-16) +884206944*a(n-17) -115250048*a(n-18) +6527616*a(n-19).
EXAMPLE
Some solutions for n=4:
..2..1..0..1..2..1..0..1....0..1..0..1..2..1..0..1....1..2..1..0..2..1..2..1
..1..2..1..0..1..0..1..0....1..0..1..0..1..2..1..2....2..1..0..1..0..2..0..2
..2..0..2..1..2..1..2..1....2..1..2..1..2..1..2..1....1..2..1..2..1..0..2..0
..0..2..0..2..0..2..1..2....1..0..1..2..1..0..1..2....2..0..2..1..2..1..0..1
..1..0..2..0..1..0..2..1....0..2..0..1..0..1..2..1....0..2..0..2..0..2..1..2
Search completed in 0.008 seconds
| 