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 A253113 Number of (n+2)X(2+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. 1

%I

%S 272,920,6334,28600,139760,502272,2259097,5670421,21399557,39120366,

%T 120865988,183106294,479657064,649963111,1500762543,1895383751,

%U 3973980189,4787211558,9308563998,10844855216,19852566186,22566185885

%N Number of (n+2)X(2+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 2 of A253119.

%H R. H. Hardin, <a href="/A253113/b253113.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>29.

%F Empirical for n mod 2 = 0: a(n) = (8/45)*n^8 + (1504/315)*n^7 + (3571/45)*n^6 - (84011/45)*n^5 + (5095469/1440)*n^4 + (27027629/360)*n^3 - (17188069/180)*n^2 - (433790313/140)*n + 11000595 for n>12.

%F Empirical for n mod 2 = 1: a(n) = (8/45)*n^8 + (1952/315)*n^7 + (301/3)*n^6 - (70897/45)*n^5 - (1084059/160)*n^4 + (23349461/180)*n^3 - (6792595/144)*n^2 - (509001211/140)*n + (327336053/32) for n>12.

%e Some solutions for n=4:

%e ..0..2..2..3....0..2..2..3....0..2..2..3....0..2..1..3....0..2..2..3

%e ..2..2..1..2....3..3..1..1....2..3..1..1....1..2..1..1....2..3..1..2

%e ..2..1..3..2....2..1..3..2....2..1..3..2....2..1..2..2....1..1..2..3

%e ..2..2..2..2....2..2..2..1....2..1..2..2....2..1..2..2....2..2..2..1

%e ..2..2..2..2....1..2..2..2....2..2..1..2....1..2..2..1....2..2..2..2

%e ..2..2..3..3....3..2..2..3....2..2..2..3....2..2..2..2....2..2..2..3

%e Knight distance matrix for n=4:

%e ..0..3..2..5

%e ..3..4..1..2

%e ..2..1..4..3

%e ..3..2..3..2

%e ..2..3..2..3

%e ..3..4..3..4

%K nonn

%O 1,1

%A _R. H. Hardin_, Dec 27 2014

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Last modified November 27 05:12 EST 2022. Contains 358362 sequences. (Running on oeis4.)