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%I #7 Dec 27 2014 18:21:06
%S 4619,28600,426571,3950239,28256241,201911670,1873418602,7421334529,
%T 54446797689,133685068828,745867325532,1354111174816,5818864244979,
%U 8824640573225,30688212938784,41602065842866,122867434214623
%N Number of (n+2)X(4+2) nonnegative integer arrays with all values the knight distance from the upper left minus as much as 2, with successive minimum path knight move differences either 0 or +1, and any unreachable value zero.
%C Column 4 of A253119
%H R. H. Hardin, <a href="/A253115/b253115.txt">Table of n, a(n) for n = 1..210</a>
%F Empirical: a(n) = a(n-1) +8*a(n-2) -8*a(n-3) -28*a(n-4) +28*a(n-5) +56*a(n-6) -56*a(n-7) -70*a(n-8) +70*a(n-9) +56*a(n-10) -56*a(n-11) -28*a(n-12) +28*a(n-13) +8*a(n-14) -8*a(n-15) -a(n-16) +a(n-17) for n>37.
%F Empirical for n mod 2 = 0: a(n) = (17563648/63)*n^8 - (753270784/45)*n^7 + (4959576064/9)*n^6 - (567642929152/45)*n^5 + (1853435878336/9)*n^4 - (104612636971216/45)*n^3 + (119980082024136/7)*n^2 - (2219288775756637/30)*n + 140910866347888 for n>20.
%F Empirical for n mod 2 = 1: a(n) = (17563648/63)*n^8 - (1523449856/105)*n^7 + (19261530112/45)*n^6 - (136247502848/15)*n^5 + (1247647337408/9)*n^4 - (21746003662832/15)*n^3 + (3075715875376376/315)*n^2 - (2653723481992701/70)*n + (127386927252109/2) for n>20.
%e Some solutions for n=2:
%e ..0..2..1..2..2..2....0..2..1..2..1..2....0..2..1..2..1..2....0..1..1..1..1..1
%e ..2..3..1..1..2..2....2..2..1..1..2..2....1..2..0..1..1..2....2..2..1..1..2..2
%e ..2..0..2..2..1..2....1..1..2..2..2..2....1..0..2..1..1..1....1..0..2..2..1..1
%e ..3..2..2..1..2..3....3..2..2..2..1..3....3..1..2..0..2..2....3..1..2..1..2..2
%e Knight distance matrix for n=2:
%e ..0..3..2..3..2..3
%e ..3..4..1..2..3..4
%e ..2..1..4..3..2..3
%e ..5..2..3..2..3..4
%K nonn
%O 1,1
%A _R. H. Hardin_, Dec 27 2014
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