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A253335
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Number of (n+2)X(1+2) nonnegative integer arrays with all values the knight distance from the upper left minus as much as 3, with successive minimum path knight move differences either 0 or +1, and any unreachable value zero.
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1
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69, 488, 1928, 7494, 27015, 87621, 319172, 945613, 2874539, 6935762, 22983454, 49969332, 119548267, 214145613, 622017082, 1069718995, 2070309209, 3028231177, 7700984072, 11401562937, 19017598555, 24702675207, 56114922678, 75683326471
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OFFSET
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1,1
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COMMENTS
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LINKS
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FORMULA
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Empirical: a(n) = a(n-1) +12*a(n-4) -12*a(n-5) -66*a(n-8) +66*a(n-9) +220*a(n-12) -220*a(n-13) -495*a(n-16) +495*a(n-17) +792*a(n-20) -792*a(n-21) -924*a(n-24) +924*a(n-25) +792*a(n-28) -792*a(n-29) -495*a(n-32) +495*a(n-33) +220*a(n-36) -220*a(n-37) -66*a(n-40) +66*a(n-41) +12*a(n-44) -12*a(n-45) -a(n-48) +a(n-49) for n>66
Empirical for n mod 4 = 0: a(n) = (1/34214400)*n^12 + (19/3326400)*n^11 + (11413/21772800)*n^10 + (2543/120960)*n^9 + (171211/1036800)*n^8 - (17681/1800)*n^7 - (364775123/3110400)*n^6 + (503575579/120960)*n^5 - (23104979617/5443200)*n^4 - (42603595499/75600)*n^3 + (271867785329/59400)*n^2 - (1488535171/330)*n - 43423533 for n>17
Empirical for n mod 4 = 1: a(n) = (1/34214400)*n^12 + (19/3326400)*n^11 + (11329/21772800)*n^10 + (139/6720)*n^9 + (1113277/7257600)*n^8 - (80513/8400)*n^7 - (321868919/3110400)*n^6 + (24501551/6048)*n^5 - (2717152987/340200)*n^4 - (13345114753/25200)*n^3 + (32592081480029/6652800)*n^2 - (327571973557/36960)*n - (2047419033/64) for n>17
Empirical for n mod 4 = 2: a(n) = (1/34214400)*n^12 + (107/19958400)*n^11 + (1453/3110400)*n^10 + (12203/725760)*n^9 + (59611/1036800)*n^8 - (2635259/302400)*n^7 - (138251441/3110400)*n^6 + (2772874619/725760)*n^5 - (630113333/24300)*n^4 - (150767030627/453600)*n^3 + (1190430331397/237600)*n^2 - (972074050223/55440)*n - (25910239/4) for n>17
Empirical for n mod 4 = 3: a(n) = (1/34214400)*n^12 + (11/1814400)*n^11 + (12403/21772800)*n^10 + (3487/145152)*n^9 + (1620937/7257600)*n^8 - (1082489/100800)*n^7 - (453300833/3110400)*n^6 + (3176992123/725760)*n^5 + (6382948247/1360800)*n^4 - (646363851967/907200)*n^3 + (35282024010953/6652800)*n^2 - (80650610293/20160)*n - (3453848023/64) for n>17
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EXAMPLE
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Some solutions for n=4
..0..2..1....0..3..2....0..1..1....0..1..1....0..1..0....0..2..1....0..2..2
..2..2..1....2..4..1....2..2..1....2..2..0....1..2..0....1..3..0....2..1..1
..1..1..2....2..1..3....1..1..2....1..0..2....0..0..1....1..1..2....1..1..2
..2..1..2....3..2..3....2..2..2....1..1..1....1..0..1....2..1..2....1..1..1
..1..1..1....2..2..2....1..2..2....0..2..0....0..1..0....1..2..1....1..2..1
..2..3..2....3..3..2....3..2..2....1..2..1....0..2..0....2..2..2....2..1..1
Knight distance matrix for n=4
..0..3..2
..3..4..1
..2..1..4
..3..2..3
..2..3..2
..3..4..3
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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