| Displaying 1-9 of 9 results found.
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T(n,k)=Number of nXk 0..1 arrays with no element equal to more than two of its horizontal, diagonal or antidiagonal neighbors and with new values introduced in order 0 sequentially upwards.
+10
13
1, 2, 2, 4, 8, 4, 8, 17, 25, 8, 16, 37, 24, 81, 16, 32, 78, 39, 60, 264, 32, 64, 169, 57, 96, 133, 857, 64, 128, 361, 87, 130, 207, 283, 2785, 128, 256, 778, 145, 176, 278, 405, 634, 9050, 256, 512, 1673, 241, 260, 358, 534, 897, 1419, 29407, 512, 1024, 3605, 397, 406
COMMENTS
Table starts
...1.....2....4....8....16....32....64...128...256...512..1024..2048..4096
...2.....8...17...37....78...169...361...778..1673..3605..7774.16777.36241
...4....25...24...39....57....87...145...241...397...669..1133..1909..3229
...8....81...60...96...130...176...260...406...636...996..1586..2552..4108
..16...264..133..207...278...358...472...658...974..1476..2246..3446..5334
..32...857..283..405...534...706...924..1190..1602..2290..3374..5036..7604
..64..2785..634..897..1106..1422..1887..2429..3093..4063..5629..8037.11711
.128..9050.1419.1975..2362..2858..3681..4795..6109..7699..9985.13483.18797
.256.29407.3092.4154..4901..5793..7126..9058.11794.14920.18660.23832.31694
.512.95557.6849.8884.10240.11986.14363.17511.22411.28951.36521.45125.56983
FORMULA
Empirical for column k:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 3*a(n-1) +a(n-2) -2*a(n-4)
k=3: a(n) = 2*a(n-1) +3*a(n-3) -3*a(n-4) -4*a(n-5) +4*a(n-6) -4*a(n-7) for n>9
k=4: a(n) = a(n-1) +a(n-2) +4*a(n-3) +a(n-4) -4*a(n-5) -5*a(n-6) +3*a(n-8) for n>10
k=5: [order 10] for n>15
k=6: [order 11] for n>17
k=7: [order 12] for n>19
Empirical for row n:
n=1: a(n) = 2*a(n-1)
n=2: a(n) = 3*a(n-1) +a(n-2) -6*a(n-3) -2*a(n-4) +4*a(n-5) for n>6
n=3: a(n) = 2*a(n-1) -a(n-2) +2*a(n-3) -2*a(n-4) for n>8
n=4: a(n) = 2*a(n-1) -a(n-2) +a(n-3) -a(n-5) for n>9
n=5: a(n) = 2*a(n-1) -a(n-2) +a(n-3) -a(n-4) +a(n-5) -a(n-6) for n>11
n=6: a(n) = 2*a(n-1) -a(n-2) +a(n-3) -a(n-4) +a(n-6) -a(n-7) for n>12
n=7: a(n) = 2*a(n-1) -a(n-2) +a(n-3) -a(n-4) +a(n-7) -a(n-8) for n>16
EXAMPLE
Some solutions for n=4 k=4
..0..1..0..0. .0..1..0..1. .0..0..1..0. .0..1..0..1. .0..1..1..0
..1..1..0..1. .0..1..0..1. .1..1..0..1. .0..1..0..1. .0..1..0..0
..1..0..0..1. .1..1..0..1. .0..1..0..1. .0..1..0..1. .0..1..0..1
..1..0..1..1. .1..0..0..0. .1..1..0..1. .1..0..1..0. .1..1..0..1
T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 2 or 4 horizontally, diagonally or antidiagonally adjacent elements, with upper left element zero.
+10
13
1, 2, 2, 4, 8, 4, 8, 20, 25, 8, 16, 52, 65, 81, 16, 32, 136, 177, 281, 264, 32, 64, 360, 519, 953, 1144, 857, 64, 128, 960, 1528, 3559, 4525, 4321, 2785, 128, 256, 2576, 4509, 14022, 20686, 19734, 18093, 9050, 256, 512, 6944, 13329, 55727, 104932, 110035
COMMENTS
Table starts
...1.....2......4.......8.......16........32.........64.........128
...2.....8.....20......52......136.......360........960........2576
...4....25.....65.....177......519......1528.......4509.......13329
...8....81....281.....953.....3559.....14022......55727......218945
..16...264...1144....4525....20686....104932.....529381.....2593409
..32...857...4321...19734...110035....708572....4521584....27478634
..64..2785..18093...96956...659949...5542827...45861206...349722102
.128..9050..72746..458537..3834162..41843808..444688101..4252740185
.256.29407.285199.2098489.21619209.305929316.4176914829.49936266116
FORMULA
Empirical for column k:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 3*a(n-1) +a(n-2) -2*a(n-4)
k=3: [order 14]
k=4: [order 48] for n>49
Empirical for row n:
n=1: a(n) = 2*a(n-1)
n=2: a(n) = 4*a(n-1) -2*a(n-2) -4*a(n-3) for n>4
n=3: [order 14] for n>15
n=4: [order 57] for n>58
EXAMPLE
Some solutions for n=5 k=4
..0..0..1..1. .0..1..0..0. .0..1..1..0. .0..0..1..1. .0..1..0..0
..1..0..0..1. .0..1..1..0. .0..0..1..0. .1..0..0..1. .1..0..1..1
..0..1..0..1. .0..0..0..0. .1..1..1..0. .1..0..1..0. .1..0..1..0
..1..1..0..1. .1..0..0..1. .0..0..1..0. .0..1..1..0. .0..1..1..0
..0..0..1..0. .1..0..0..1. .1..0..1..1. .0..1..1..0. .1..0..0..1
T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 2 or 5 horizontally, diagonally or antidiagonally adjacent elements, with upper left element zero.
+10
12
1, 2, 2, 4, 8, 4, 8, 17, 25, 8, 16, 37, 30, 81, 16, 32, 78, 66, 120, 264, 32, 64, 169, 142, 259, 441, 857, 64, 128, 361, 289, 698, 994, 1431, 2785, 128, 256, 778, 599, 1650, 3553, 3548, 4754, 9050, 256, 512, 1673, 1268, 3655, 11998, 16123, 12442, 16423, 29407, 512, 1024
COMMENTS
Table starts
...1.....2.....4......8......16.......32........64........128.........256
...2.....8....17.....37......78......169.......361........778........1673
...4....25....30.....66.....142......289.......599.......1268........2660
...8....81...120....259.....698.....1650......3655.......8544.......20246
..16...264...441....994....3553....11998.....33596......98387......295578
..32...857..1431...3548...16123....84581....306064....1137414.....4669030
..64..2785..4754..12442...69537...518736...2342685...10963076....58365289
.128..9050.16423..46066..318944..3464956..20184979..119713006...845708279
.256.29407.55848.166341.1443006.23154338.170625048.1280996808.12137387233
FORMULA
Empirical for column k:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 3*a(n-1) +a(n-2) -2*a(n-4)
k=3: [order 13]
k=4: [order 31] for n>33
k=5: [order 66] for n>73
Empirical for row n:
n=1: a(n) = 2*a(n-1)
n=2: a(n) = 3*a(n-1) +a(n-2) -6*a(n-3) -2*a(n-4) +4*a(n-5) for n>6
n=3: a(n) = 3*a(n-1) -3*a(n-2) +4*a(n-3) -4*a(n-4) +a(n-5) for n>9
n=4: [order 16] for n>21
n=5: [order 31] for n>39
n=6: [order 79] for n>86
EXAMPLE
Some solutions for n=5 k=4
..0..0..1..0. .0..1..0..1. .0..1..0..1. .0..0..1..1. .0..1..0..1
..1..0..0..1. .0..0..0..1. .0..0..1..1. .0..0..1..0. .1..0..1..0
..1..0..1..0. .1..1..0..1. .0..1..0..1. .1..1..1..0. .1..0..1..0
..1..1..1..0. .0..1..1..0. .0..1..1..1. .1..0..1..0. .1..0..1..0
..0..0..1..0. .0..1..0..1. .1..1..0..0. .1..0..1..1. .0..1..0..1
T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 2 or 6 horizontally, diagonally or antidiagonally adjacent elements, with upper left element zero.
+10
12
1, 2, 2, 4, 8, 4, 8, 17, 25, 8, 16, 37, 25, 81, 16, 32, 78, 45, 71, 264, 32, 64, 169, 79, 130, 191, 857, 64, 128, 361, 146, 251, 367, 498, 2785, 128, 256, 778, 286, 497, 896, 917, 1321, 9050, 256, 512, 1673, 563, 1051, 2229, 2669, 2533, 3505, 29407, 512, 1024, 3605
COMMENTS
Table starts
...1.....2....4.....8....16.....32......64......128.......256........512
...2.....8...17....37....78....169.....361......778......1673.......3605
...4....25...25....45....79....146.....286......563......1114.......2222
...8....81...71...130...251....497....1051.....2274......4999......10972
..16...264..191...367...896...2229....5771....15303.....41564.....113363
..32...857..498...917..2669...9015...30536...106951....390471....1418770
..64..2785.1321..2533..8283..36510..159831...722300...3485735...16654024
.128..9050.3505..6871.25770.150072..851266..5004858..32283797..204853561
.256.29407.9240.18334.79047.606756.4471550.34111427.294788397.2490596189
FORMULA
Empirical for column k:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 3*a(n-1) +a(n-2) -2*a(n-4)
k=3: [order 9] for n>11
k=4: [order 8] for n>10
k=5: [order 15] for n>20
k=6: [order 21] for n>27
k=7: [order 35] for n>43
Empirical for row n:
n=1: a(n) = 2*a(n-1)
n=2: a(n) = 3*a(n-1) +a(n-2) -6*a(n-3) -2*a(n-4) +4*a(n-5) for n>6
n=3: a(n) = 2*a(n-1) +a(n-3) -2*a(n-4) for n>8
n=4: [order 8] for n>11
n=5: [order 16] for n>21
n=6: [order 29] for n>35
n=7: [order 51] for n>62
EXAMPLE
Some solutions for n=5 k=4
..0..0..1..1. .0..1..1..0. .0..1..0..1. .0..0..1..0. .0..1..0..0
..1..0..1..0. .0..0..1..0. .0..1..0..1. .1..0..1..0. .1..1..0..1
..1..0..1..0. .1..0..1..0. .0..1..1..1. .1..0..1..0. .1..0..0..1
..0..1..0..1. .0..0..1..1. .0..1..0..1. .1..1..1..0. .1..0..1..1
..0..1..0..1. .1..1..0..0. .0..1..0..1. .1..0..1..1. .0..0..1..0
T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 2, 4 or 5 horizontally, diagonally or antidiagonally adjacent elements, with upper left element zero.
+10
12
1, 2, 2, 4, 8, 4, 8, 20, 25, 8, 16, 52, 75, 81, 16, 32, 136, 220, 389, 264, 32, 64, 360, 731, 1494, 1852, 857, 64, 128, 960, 2419, 6905, 9050, 8485, 2785, 128, 256, 2576, 7900, 34957, 57684, 51476, 41247, 9050, 256, 512, 6944, 25645, 170262, 455224, 454733
COMMENTS
Table starts
...1.....2......4........8........16..........32...........64............128
...2.....8.....20.......52.......136.........360..........960...........2576
...4....25.....75......220.......731........2419.........7900..........25645
...8....81....389.....1494......6905.......34957.......170262.........800847
..16...264...1852.....9050.....57684......455224......3252934.......21373833
..32...857...8485....51476....454733.....5520999.....58122740......535479917
..64..2785..41247...317216...3825201....72576189...1135257814....14619284475
.128..9050.196946..1927184..32123578...960564532..22294687742...402121386052
.256.29407.931243.11448173.264639685.12417289008.427140472935.10807066934912
FORMULA
Empirical for column k:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 3*a(n-1) +a(n-2) -2*a(n-4)
k=3: [order 14]
k=4: [order 49] for n>51
Empirical for row n:
n=1: a(n) = 2*a(n-1)
n=2: a(n) = 4*a(n-1) -2*a(n-2) -4*a(n-3) for n>4
n=3: [order 15] for n>16
n=4: [order 67] for n>68
EXAMPLE
Some solutions for n=5 k=4
..0..0..1..0. .0..1..0..1. .0..1..0..1. .0..1..1..0. .0..1..0..1
..0..1..1..1. .1..1..1..1. .1..0..1..0. .1..0..0..0. .0..0..0..1
..0..1..0..1. .1..0..1..1. .1..1..1..1. .1..1..1..0. .1..0..1..1
..0..0..0..1. .1..1..0..1. .0..1..0..1. .1..0..1..0. .1..1..1..1
..1..1..1..0. .1..1..0..1. .0..1..0..0. .0..0..0..0. .0..1..0..0
T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 2, 4 or 6 horizontally, diagonally or antidiagonally adjacent elements, with upper left element zero.
+10
12
1, 2, 2, 4, 8, 4, 8, 20, 25, 8, 16, 52, 68, 81, 16, 32, 136, 187, 308, 264, 32, 64, 360, 579, 1047, 1320, 857, 64, 128, 960, 1797, 4237, 5299, 5220, 2785, 128, 256, 2576, 5571, 18513, 27719, 24030, 22652, 9050, 256, 512, 6944, 17382, 79945, 166978, 160253
COMMENTS
Table starts
...1.....2......4.......8.......16........32..........64..........128
...2.....8.....20......52......136.......360.........960.........2576
...4....25.....68.....187......579......1797........5571........17382
...8....81....308....1047.....4237.....18513.......79945.......344190
..16...264...1320....5299....27719....166978......970892......5570473
..32...857...5220...24030...160253...1298140.....9976891.....75234550
..64..2785..22652..123538..1044810..11644063...121524556...1219773362
.128..9050..95220..612923..6647878.101648313..1423906011..18974341664
.256.29407.390580.2935811.40950485.858924912.16163776349.285987277414
FORMULA
Empirical for column k:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 3*a(n-1) +a(n-2) -2*a(n-4)
k=3: [order 10]
k=4: [order 43] for n>44
Empirical for row n:
n=1: a(n) = 2*a(n-1)
n=2: a(n) = 4*a(n-1) -2*a(n-2) -4*a(n-3) for n>4
n=3: [order 15] for n>16
n=4: [order 62] for n>64
EXAMPLE
Some solutions for n=5 k=4
..0..0..0..1. .0..0..0..1. .0..1..0..1. .0..1..1..0. .0..1..0..1
..0..1..0..1. .1..1..1..1. .1..1..0..1. .0..0..1..1. .0..0..0..1
..0..1..0..0. .0..0..0..1. .0..0..0..1. .0..0..0..0. .1..0..1..1
..0..1..1..0. .1..1..1..1. .1..1..0..1. .1..1..0..0. .0..1..0..1
..0..1..0..1. .0..1..0..0. .1..1..0..1. .0..1..1..0. .0..1..0..1
T(n,k) = Number of n X k 0..1 arrays with every element equal to 0, 1, 2, 5 or 6 horizontally, diagonally or antidiagonally adjacent elements, with upper left element zero.
+10
12
1, 2, 2, 4, 8, 4, 8, 17, 25, 8, 16, 37, 31, 81, 16, 32, 78, 72, 135, 264, 32, 64, 169, 170, 297, 547, 857, 64, 128, 361, 382, 931, 1302, 1938, 2785, 128, 256, 778, 873, 2569, 5613, 5050, 6965, 9050, 256, 512, 1673, 2013, 6668, 24751, 29124, 20188, 25953, 29407, 512
COMMENTS
Table starts
...1.....2.....4......8......16........32.........64.........128..........256
...2.....8....17.....37......78.......169........361.........778.........1673
...4....25....31.....72.....170.......382........873........2013.........4618
...8....81...135....297.....931......2569.......6668.......17934........48674
..16...264...547...1302....5613.....24751......88153......325770......1253442
..32...857..1938...5050...29124....218980....1064399.....5463686.....31140302
..64..2785..6965..20188..148501...1803435...11901719....82524101....673749714
.128..9050.25953..81926..786247..15559135..141429948..1341217380..15951682785
.256.29407.95602.329957.4126586.133939941.1665576584.21624053966.375045359077
FORMULA
Empirical for column k:
k=1: a(n) = 2*a(n-1).
k=2: a(n) = 3*a(n-1) +a(n-2) -2*a(n-4).
k=3: [order 13].
k=4: [order 31] for n>33.
k=5: [order 66] for n>73.
Empirical for row n:
n=1: a(n) = 2*a(n-1).
n=2: a(n) = 3*a(n-1) +a(n-2) -6*a(n-3) -2*a(n-4) +4*a(n-5) for n>6.
n=3: a(n) = 3*a(n-1) -2*a(n-2) +2*a(n-3) -3*a(n-4) +a(n-5) for n>9.
n=4: [order 17] for n>22.
n=5: [order 34] for n>40.
n=6: [order 84] for n>90.
EXAMPLE
Some solutions for n=5, k=4
..0..1..0..1. .0..0..1..0. .0..1..0..1. .0..1..0..1. .0..0..1..1
..0..1..0..1. .0..1..1..0. .1..0..0..1. .0..0..0..1. .1..0..1..0
..0..1..1..1. .0..1..0..0. .0..1..0..1. .0..1..1..1. .1..0..0..0
..0..1..0..0. .0..1..0..1. .0..0..0..1. .0..1..0..1. .1..0..1..1
..0..1..0..1. .0..1..0..0. .1..1..0..1. .0..1..0..0. .1..0..1..0
T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 2, 4, 5 or 6 horizontally, diagonally or antidiagonally adjacent elements, with upper left element zero.
+10
12
1, 2, 2, 4, 8, 4, 8, 20, 25, 8, 16, 52, 78, 81, 16, 32, 136, 230, 420, 264, 32, 64, 360, 801, 1596, 2088, 857, 64, 128, 960, 2764, 7951, 10128, 9916, 2785, 128, 256, 2576, 9367, 43448, 72890, 59086, 49740, 9050, 256, 512, 6944, 31703, 222205, 656838, 613788
COMMENTS
Table starts
...1.....2.......4........8........16..........32............64............128
...2.....8......20.......52.......136.........360...........960...........2576
...4....25......78......230.......801........2764..........9367..........31703
...8....81.....420.....1596......7951.......43448........222205........1111647
..16...264....2088....10128.....72890......656838.......5122419.......37600463
..32...857....9916....59086....613788.....8883058.....104285205.....1111578890
..64..2785...49740...377020...5520182...130480638....2346396384....36337215370
.128..9050..245904..2366996..49761178..1939853204...53305999504..1203039901394
.256.29407.1204720.14512973.438900203.28081434251.1176790967846.38683326206589
FORMULA
Empirical for column k:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 3*a(n-1) +a(n-2) -2*a(n-4)
k=3: [order 14]
k=4: [order 49] for n>51
Empirical for row n:
n=1: a(n) = 2*a(n-1)
n=2: a(n) = 4*a(n-1) -2*a(n-2) -4*a(n-3) for n>4
n=3: [order 15] for n>16
n=4: [order 63] for n>65
EXAMPLE
Some solutions for n=5 k=4
..0..0..1..0. .0..0..1..0. .0..0..0..0. .0..1..0..0. .0..1..0..1
..0..1..1..0. .1..0..1..0. .1..0..1..0. .0..1..0..1. .0..0..1..0
..1..1..1..0. .1..1..1..0. .1..1..1..0. .1..0..0..1. .0..1..0..1
..1..0..1..0. .0..0..1..1. .1..0..0..0. .0..1..1..1. .0..1..1..1
..1..0..0..0. .1..1..0..1. .1..0..1..0. .0..1..0..1. .0..1..0..0
T(n,k)=Number of nXk 0..1 arrays with no element equal to exactly three horizontal or vertical neighbors, with new values 0..1 introduced in row major order
+10
7
1, 2, 2, 4, 8, 4, 8, 25, 25, 8, 16, 81, 129, 81, 16, 32, 264, 637, 637, 264, 32, 64, 857, 3147, 4989, 3147, 857, 64, 128, 2785, 15590, 38634, 38634, 15590, 2785, 128, 256, 9050, 77269, 299329, 470170, 299329, 77269, 9050, 256, 512, 29407, 382933, 2319211
COMMENTS
Table starts
...1.....2.......4..........8...........16.............32...............64
...2.....8......25.........81..........264............857.............2785
...4....25.....129........637.........3147..........15590............77269
...8....81.....637.......4989........38634.........299329..........2319211
..16...264....3147......38634.......470170........5723270.........69637692
..32...857...15590.....299329......5723270......109584463.......2097636322
..64..2785...77269....2319211.....69637692.....2097636322......63213422873
.128..9050..382933...17971509....847232955....40145256595....1904782876182
.256.29407.1897681..139264013..10307784448...768275538054...57392644533582
.512.95557.9404215.1079178635.125409399932.14702668840259.1729241310772922
FORMULA
Empirical for column k:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 3*a(n-1) +a(n-2) -2*a(n-4)
k=3: a(n) = 4*a(n-1) +4*a(n-2) +2*a(n-3) +7*a(n-4) +6*a(n-5) -a(n-6) -a(n-7) -a(n-8)
k=4: [order 32]
EXAMPLE
Some solutions for n=4 k=4
..0..1..1..0....0..0..1..0....0..1..1..1....0..0..1..0....0..0..0..1
..0..1..0..0....0..1..0..1....1..1..0..0....1..0..1..0....1..1..0..1
..1..0..1..1....1..0..0..1....1..0..0..0....1..0..0..1....0..0..1..0
..1..0..0..1....1..0..1..0....1..0..0..1....1..1..0..0....1..0..0..1
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