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Number of nX3 0..1 arrays with every element unequal to 0, 1, 2, 6, 7 or 8 king-move adjacent elements, with upper left element zero.
+10
1
4, 3, 8, 9, 21, 42, 81, 165, 330, 657, 1317, 2634, 5265, 10533, 21066, 42129, 84261, 168522, 337041, 674085, 1348170, 2696337, 5392677, 10785354, 21570705, 43141413, 86282826, 172565649, 345131301, 690262602, 1380525201, 2761050405, 5522100810
FORMULA
Empirical: a(n) = a(n-1) +a(n-2) +2*a(n-3) for n>6
EXAMPLE
Some solutions for n=5
..0..0..0. .0..0..0. .0..0..0. .0..0..0. .0..0..0. .0..0..0. .0..0..0
..0..0..0. .0..0..0. .0..1..0. .0..0..0. .0..1..0. .0..0..0. .0..0..0
..0..1..0. .0..1..0. .1..0..1. .0..0..0. .0..0..0. .0..1..0. .0..0..0
..0..0..0. .0..1..0. .1..1..1. .0..1..0. .0..1..0. .1..0..1. .0..1..0
..0..0..0. .0..0..0. .1..1..1. .1..0..0. .1..0..0. .1..1..1. .0..0..1
Number of nX4 0..1 arrays with every element unequal to 0, 1, 2, 6, 7 or 8 king-move adjacent elements, with upper left element zero.
+10
1
8, 5, 9, 23, 44, 125, 313, 773, 1964, 4948, 12456, 31393, 79085, 199281, 502100, 1265024, 3187400, 8030877, 20234253, 50981805, 128452324, 323644732, 815446408, 2054575929, 5176652093, 13042948233, 32862647844, 82799810200, 208620089912
FORMULA
Empirical: a(n) = a(n-1) +2*a(n-2) +5*a(n-3) +a(n-4) -2*a(n-5) -6*a(n-6) -4*a(n-7) for n>11
EXAMPLE
Some solutions for n=5
..0..0..0..0. .0..0..0..0. .0..0..0..0. .0..0..0..0. .0..0..0..0
..0..1..0..0. .0..0..1..0. .0..0..0..0. .0..0..0..0. .0..1..1..0
..0..0..0..0. .0..0..0..0. .0..0..0..0. .0..1..0..0. .0..0..0..0
..0..0..1..0. .0..1..0..0. .0..1..0..0. .0..1..0..0. .0..0..0..0
..0..0..0..0. .0..0..0..0. .0..0..0..0. .0..0..0..0. .0..0..0..0
Number of nX5 0..1 arrays with every element unequal to 0, 1, 2, 6, 7 or 8 king-move adjacent elements, with upper left element zero.
+10
1
16, 8, 21, 44, 88, 359, 1125, 3117, 10022, 31545, 95424, 296211, 920485, 2836189, 8769823, 27145969, 83899059, 259392614, 802225003, 2480455071, 7669523490, 23715722335, 73331519730, 226746858798, 701128606584, 2167967516063
FORMULA
Empirical: a(n) = a(n-1) +3*a(n-2) +12*a(n-3) +2*a(n-4) -9*a(n-5) -27*a(n-6) -6*a(n-7) for n>11
EXAMPLE
Some solutions for n=5
..0..1..1..1..1. .0..0..0..0..0. .0..0..0..0..0. .0..0..0..0..0
..1..0..1..1..1. .0..0..0..0..0. .0..0..0..0..0. .0..1..0..0..0
..1..1..1..0..1. .0..0..0..0..0. .0..0..0..1..0. .0..0..1..0..0
..1..1..1..1..1. .0..0..1..0..0. .0..0..1..0..0. .0..0..0..1..0
..1..1..1..1..1. .0..0..0..0..0. .0..0..0..0..0. .0..0..0..0..0
Number of nX6 0..1 arrays with every element unequal to 0, 1, 2, 6, 7 or 8 king-move adjacent elements, with upper left element zero.
+10
1
32, 13, 42, 125, 359, 2081, 9442, 38020, 174483, 785604, 3427907, 15257880, 67982637, 300964560, 1335669886, 5933105093, 26325240756, 116828769988, 518611346889, 2301753155715, 10215798261389, 45343030343793
FORMULA
Empirical: a(n) = 2*a(n-1) +5*a(n-2) +33*a(n-3) -9*a(n-4) -53*a(n-5) -229*a(n-6) +42*a(n-7) +42*a(n-8) +386*a(n-9) -47*a(n-10) -52*a(n-11) -288*a(n-12) +62*a(n-13) +216*a(n-15) +84*a(n-16) for n>21
EXAMPLE
Some solutions for n=5
..0..0..0..0..0..1. .0..0..0..0..0..0. .0..0..0..0..0..0. .0..0..0..0..0..0
..0..0..0..0..1..0. .0..0..0..0..0..0. .0..0..0..1..1..0. .0..0..1..0..1..0
..0..0..0..0..0..0. .0..0..0..1..0..0. .0..0..0..0..0..0. .0..1..0..0..0..0
..0..0..0..0..1..0. .0..0..0..0..1..0. .0..1..0..0..0..0. .0..0..0..0..0..0
..0..0..0..0..0..0. .0..0..0..0..0..1. .0..0..0..0..0..0. .0..0..0..0..0..0
Number of n X 7 0..1 arrays with every element unequal to 0, 1, 2, 6, 7 or 8 king-move adjacent elements, with upper left element zero.
+10
1
64, 21, 81, 313, 1125, 9442, 61454, 342508, 2212378, 14109348, 86477928, 541484462, 3400774549, 21195268549, 132406833468, 828277255783, 5174739183171, 32333305874846, 202097767759162, 1262976862661335, 7892519184607337
FORMULA
Empirical: a(n) = 2*a(n-1) +12*a(n-2) +110*a(n-3) +26*a(n-4) -415*a(n-5) -3065*a(n-6) -1514*a(n-7) +3667*a(n-8) +33884*a(n-9) +19541*a(n-10) -12055*a(n-11) -215062*a(n-12) -98825*a(n-13) +502*a(n-14) +882896*a(n-15) +306568*a(n-16) +150206*a(n-17) -2481993*a(n-18) -650282*a(n-19) -864282*a(n-20) +4763862*a(n-21) +1085395*a(n-22) +2069629*a(n-23) -5956995*a(n-24) -1394383*a(n-25) -2946869*a(n-26) +5150227*a(n-27) +1230622*a(n-28) +2722066*a(n-29) -3058411*a(n-30) -1045031*a(n-31) -1731070*a(n-32) +1264814*a(n-33) +738227*a(n-34) +812157*a(n-35) -315914*a(n-36) -356366*a(n-37) -245556*a(n-38) +10608*a(n-39) +98176*a(n-40) +36476*a(n-41) +7552*a(n-42) -8952*a(n-43) -1648*a(n-44) -256*a(n-45) +352*a(n-46) for n>52.
EXAMPLE
Some solutions for n=5
..0..0..0..0..0..0..1. .0..0..0..0..0..0..0. .0..0..0..0..0..0..0
..0..1..0..0..0..1..0. .0..0..1..0..0..0..0. .0..0..1..0..0..1..0
..0..0..1..0..0..0..0. .0..0..0..0..0..1..0. .0..0..0..0..0..0..0
..0..0..0..0..0..0..0. .0..0..0..1..0..0..0. .0..0..1..0..0..0..0
..0..0..0..0..0..0..0. .0..0..0..0..0..0..0. .0..0..0..0..0..0..0
Number of nXn 0..1 arrays with every element unequal to 0, 1, 2, 6, 7 or 8 king-move adjacent elements, with upper left element zero.
+10
0
1, 4, 8, 23, 88, 2081, 61454, 2431114, 257262996, 52155197669, 17033276992389, 11335703888987492, 14840834772251861064, 35171617032782262638585
EXAMPLE
Some solutions for n=5
..0..1..1..1..1. .0..0..0..0..0. .0..0..0..0..0. .0..0..0..0..0
..1..0..1..1..1. .0..0..0..0..0. .0..0..0..0..0. .0..0..0..0..0
..1..1..1..1..1. .0..0..0..0..0. .0..0..0..0..0. .0..0..0..0..0
..1..1..1..0..1. .0..0..1..1..0. .0..1..1..0..0. .0..1..0..1..0
..1..1..1..1..0. .0..0..0..0..0. .0..0..0..0..0. .1..0..0..0..0
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