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A253338
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Number of (n+2)X(4+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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7494, 59519, 722826, 7184212, 65795210, 621408160, 6203113283, 46327921378, 431150608201, 2513803675142, 21694110698916, 91728436841790, 701214300325482, 2062603211477260, 13146712894951064, 29023341800426220
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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>55.
Empirical for n mod 2 = 0: a(n) = (8992587776/93555)*n^12 - (1754527694848/155925)*n^11 + (4454499745792/6075)*n^10 - (97850932330496/2835)*n^9 + (503920339091456/405)*n^8 - (163857161734094848/4725)*n^7 + (31746767654773668352/42525)*n^6 - (34889484014881237376/2835)*n^5 + (184886920914132728164/1215)*n^4 - (19246032261612188624813/14175)*n^3 + (485106832651191238006429/59400)*n^2 - (403481645478857418044233/13860)*n + 44942549657886439191 for n>30.
Empirical for n mod 2 = 1: a(n) = (8992587776/93555)*n^12 - (524891979776/51975)*n^11 + (734730911744/1215)*n^10 - (75896779177984/2835)*n^9 + (370458004717568/405)*n^8 - (113927599393374208/4725)*n^7 + (4158011324223812096/8505)*n^6 - (7143754455560924288/945)*n^5 + (105914562844059145828/1215)*n^4 - (10178334481445614975453/14175)*n^3 + (46495471701987542783201/11880)*n^2 - (84473612843189842494151/6930)*n + (120539555998778748191/8) for n>30.
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EXAMPLE
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Some solutions for n=2
..0..1..0..1..0..2....0..2..1..2..2..1....0..2..2..3..1..3....0..1..1..1..0..1
..1..2..0..1..1..2....2..2..1..1..2..2....2..3..1..2..2..3....1..1..0..1..1..2
..1..0..1..1..1..1....2..1..2..2..1..2....2..1..3..2..2..2....0..0..2..1..1..1
..2..0..1..0..1..2....3..2..2..2..2..3....3..1..3..2..2..3....2..0..1..1..2..1
Knight distance matrix for n=2
..0..3..2..3..2..3
..3..4..1..2..3..4
..2..1..4..3..2..3
..5..2..3..2..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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