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Number of n X 2 0..2 arrays with no element unequal to a strict majority of its horizontal and vertical neighbors, with the exception of exactly one element, and with new values introduced in order 0 sequentially upwards.
+10
1
0, 4, 10, 24, 54, 116, 250, 528, 1118, 2348, 4930, 10312, 21542, 44900, 93450, 194176, 402926, 834972, 1728210, 3572920, 7378870, 15223764, 31379610, 64623344, 132974974, 273406476, 561726050, 1153278248, 2366208838, 4851722308
FORMULA
Empirical: a(n) = 3*a(n-1) + a(n-2) - 7*a(n-3) + 4*a(n-5).
Empirical g.f.: 2*x^2*(2 - x - 5*x^2) / ((1 - x)*(1 + x)^2*(1 - 2*x)^2). - Colin Barker, Feb 12 2019
EXAMPLE
Some solutions for n=4:
..0..1. .0..1. .0..0. .0..1. .0..0. .0..0. .0..0. .0..0. .0..0. .0..0
..0..0. .0..1. .0..1. .0..0. .1..1. .0..0. .0..1. .1..0. .0..0. .1..0
..0..0. .1..1. .0..0. .0..0. .1..1. .0..0. .0..1. .0..0. .0..1. .1..0
..1..1. .1..1. .0..0. .2..2. .1..2. .0..1. .0..1. .0..0. .0..0. .1..0
Number of n X 3 0..2 arrays with no element unequal to a strict majority of its horizontal and vertical neighbors, with the exception of exactly one element, and with new values introduced in order 0 sequentially upwards.
+10
1
2, 10, 49, 168, 557, 1758, 5441, 16500, 49253, 145290, 424425, 1229824, 3539405, 10127350, 28832593, 81728396, 230776757, 649427170, 1821994809, 5097729560, 14227693853, 39620451150, 110107647905, 305424435364, 845754303493
FORMULA
Empirical: a(n) = 4*a(n-1) - 2*a(n-2) - 9*a(n-4) - 4*a(n-5) - 4*a(n-6) for n>9.
Empirical g.f.: x*(2 + 2*x + 13*x^2 - 8*x^3 + x^4 - 36*x^5 + 12*x^6 + 16*x^8) / (1 - 2*x - x^2 - 2*x^3)^2. - Colin Barker, Feb 11 2019
EXAMPLE
Some solutions for n=4:
..0..0..0. .0..0..0. .0..1..1. .0..1..1. .0..0..0. .0..1..1. .0..0..1
..1..1..1. .1..1..1. .1..1..1. .0..1..1. .0..0..0. .0..1..1. .1..1..1
..1..1..1. .1..1..1. .1..1..1. .0..1..1. .1..0..0. .0..0..0. .1..1..1
..2..2..1. .1..0..1. .0..0..0. .0..0..1. .1..0..0. .0..2..0. .2..2..2
Number of nX4 0..2 arrays with no element unequal to a strict majority of its horizontal and vertical neighbors, with the exception of exactly one element, and with new values introduced in order 0 sequentially upwards.
+10
1
2, 24, 168, 972, 5200, 26632, 134898, 668668, 3278294, 15902088, 76515244, 365585432, 1736575416, 8206953072, 38615236370, 180986296644, 845359203352, 3936439186972, 18279759467004, 84675466304900, 391350188916098
FORMULA
Empirical: a(n) = 16*a(n-1) -96*a(n-2) +234*a(n-3) +74*a(n-4) -1878*a(n-5) +5053*a(n-6) -6092*a(n-7) -1433*a(n-8) +20562*a(n-9) -43095*a(n-10) +59066*a(n-11) -74667*a(n-12) +116460*a(n-13) -201487*a(n-14) +270314*a(n-15) -234247*a(n-16) +69548*a(n-17) +147085*a(n-18) -260796*a(n-19) +309393*a(n-20) -437158*a(n-21) +682254*a(n-22) -940810*a(n-23) +842002*a(n-24) -383734*a(n-25) -172142*a(n-26) +689424*a(n-27) -853200*a(n-28) +709896*a(n-29) -491572*a(n-30) +247956*a(n-31) -95569*a(n-32) +32832*a(n-33) -3312*a(n-34) -624*a(n-35) +192*a(n-36) -384*a(n-37) -64*a(n-38) for n>41
EXAMPLE
Some solutions for n=4
..0..0..1..1. .0..0..0..1. .0..0..1..1. .0..1..0..0. .0..0..0..0
..0..0..1..1. .2..2..0..0. .0..0..1..1. .0..0..0..0. .1..1..0..0
..0..2..1..1. .2..2..0..0. .1..2..2..2. .0..1..1..1. .1..1..1..0
..0..2..2..2. .2..2..2..2. .1..2..2..2. .0..1..1..1. .1..1..1..1
Number of nX5 0..2 arrays with no element unequal to a strict majority of its horizontal and vertical neighbors, with the exception of exactly one element, and with new values introduced in order 0 sequentially upwards.
+10
1
8, 54, 557, 5200, 44893, 373516, 3010179, 23836450, 185854745, 1432781380, 10940440805, 82884481996, 623764753929, 4667668549882, 34757207667097, 257707173342548, 1903562116898937, 14013650638972172, 102857118300106915
EXAMPLE
Some solutions for n=4
..0..0..1..0..0. .0..0..0..0..0. .0..0..0..0..0. .0..0..0..0..1
..0..0..1..0..0. .1..1..0..0..0. .0..0..0..0..1. .0..0..0..0..0
..2..2..1..1..0. .1..1..1..0..1. .0..0..0..2..2. .2..2..0..0..0
..2..2..1..1..0. .1..1..1..0..0. .2..2..2..2..2. .2..2..2..2..2
Number of nX6 0..2 arrays with no element unequal to a strict majority of its horizontal and vertical neighbors, with the exception of exactly one element, and with new values introduced in order 0 sequentially upwards.
+10
1
14, 116, 1758, 26632, 373516, 4989784, 64921744, 827573664, 10392951988, 129001348624, 1586027917024, 19345996306220, 234406391335134, 2824014683941452, 33854795060931456, 404112060548429912
EXAMPLE
Some solutions for n=4
..0..0..0..0..0..1. .0..0..0..0..1..1. .0..0..1..1..0..0. .0..1..1..2..2..2
..0..0..0..0..0..1. .2..2..0..0..1..2. .0..0..1..1..1..1. .0..1..1..2..2..1
..0..2..0..1..1..1. .2..2..1..1..1..2. .0..0..1..0..0..1. .0..1..1..0..0..0
..0..0..0..1..1..1. .1..1..1..1..1..2. .1..1..1..0..0..1. .0..1..1..0..0..0
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