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A253336
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Number of (n+2)X(2+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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488, 2028, 11581, 59519, 306822, 1472184, 7426426, 30066852, 142944394, 470066369, 2057470881, 5310820947, 20643033198, 42416116870, 143775300377, 248844171284, 741261828329, 1137682147050, 3032909525755, 4278092511504
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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-2) -12*a(n-3) -66*a(n-4) +66*a(n-5) +220*a(n-6) -220*a(n-7) -495*a(n-8) +495*a(n-9) +792*a(n-10) -792*a(n-11) -924*a(n-12) +924*a(n-13) +792*a(n-14) -792*a(n-15) -495*a(n-16) +495*a(n-17) +220*a(n-18) -220*a(n-19) -66*a(n-20) +66*a(n-21) +12*a(n-22) -12*a(n-23) -a(n-24) +a(n-25) for n>43.
Empirical for n mod 2 = 0: a(n) = (16/66825)*n^12 + (2624/155925)*n^11 + (809/1215)*n^10 - (21302/2835)*n^9 - (62071883/226800)*n^8 - (55181753/37800)*n^7 + (52986729011/311040)*n^6 - (272758912049/181440)*n^5 - (7934189342537/777600)*n^4 + (88549659794273/453600)*n^3 - (36155261436353/66528)*n^2 - (87347525171009/27720)*n + 13892843858 for n>18.
Empirical for n mod 2 = 1: a(n) = (16/66825)*n^12 + (1024/51975)*n^11 + (34507/42525)*n^10 - (7996/2835)*n^9 - (88305083/226800)*n^8 - (97992269/37800)*n^7 + (267135548623/1555200)*n^6 - (76565191531/120960)*n^5 - (227590352001343/10886400)*n^4 + (184332543438053/907200)*n^3 - (914930300044087/13305600)*n^2 - (2333356583608291/443520)*n + (3762414152367/256) for n>18.
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EXAMPLE
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Some solutions for n=4
..0..0..1..2....0..2..1..2....0..1..0..2....0..1..0..2....0..2..2..3
..1..2..0..0....1..2..1..2....0..1..0..0....0..1..0..0....3..2..1..2
..0..0..1..1....2..1..2..1....0..0..1..1....0..0..1..1....2..1..3..2
..1..1..1..0....1..1..2..1....1..0..0..1....1..0..1..1....2..2..2..1
..1..1..1..1....1..2..1..2....0..1..0..0....1..1..1..1....2..2..2..2
..1..2..1..1....1..2..2..2....0..1..1..1....1..2..1..1....3..3..2..2
Knight distance matrix for n=4
..0..3..2..5
..3..4..1..2
..2..1..4..3
..3..2..3..2
..2..3..2..3
..3..4..3..4
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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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