%I #9 Sep 27 2015 17:42:39

%S 17,56,257,642,1581,2389,5716,7691,11429,13229,25870,30605,39701,

%T 42941,73200,81815,99709,104837,163706,177321,208349,215813,316972,

%U 336707,386101,396349,556166,583141,657029,670509,908040,943375,1048781,1065941

%N Number of (n+2)X(1+2) nonnegative integer arrays with all values the knight distance from the upper left minus as much as 1, with successive minimum path knight move differences either 0 or +1, and any unreachable value zero.

%C Column 1 of A253424.

%H R. H. Hardin, <a href="/A253417/b253417.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = a(n-1) +4*a(n-4) -4*a(n-5) -6*a(n-8) +6*a(n-9) +4*a(n-12) -4*a(n-13) -a(n-16) +a(n-17) for n>21.

%F Empirical for n mod 4 = 0: a(n) = (7/12)*n^4 + 9*n^3 + (563/12)*n^2 - (745/2)*n + 671 for n>4.

%F Empirical for n mod 4 = 1: a(n) = (7/12)*n^4 + 9*n^3 + (479/12)*n^2 - (643/2)*n + 701 for n>4.

%F Empirical for n mod 4 = 2: a(n) = (7/12)*n^4 + (20/3)*n^3 + (365/12)*n^2 - (1058/3)*n + 1214 for n>4.

%F Empirical for n mod 4 = 3: a(n) = (7/12)*n^4 + (34/3)*n^3 + (509/12)*n^2 - (928/3)*n + 515 for n>4.

%e Some solutions for n=2

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

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

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

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

%e Knight distance matrix for n=2

%e ..0..3..2

%e ..3..4..1

%e ..2..1..4

%e ..5..2..3

%K nonn

%O 1,1

%A _R. H. Hardin_, Dec 31 2014