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A253112 Number of (n+2)X(1+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 53,272,1342,4619,14541,34786,113891,233392,525617,853971,2327441,

%T 3609337,6243433,8189113,18881723,25291483,37714007,44777219,90840301,

%U 112309445,153843349,173399285,320621751,377531207,490093227,535756183

%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 2, with successive minimum path knight move differences either 0 or +1, and any unreachable value zero.

%C Column 1 of A253119.

%H R. H. Hardin, <a href="/A253112/b253112.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = a(n-1) +8*a(n-4) -8*a(n-5) -28*a(n-8) +28*a(n-9) +56*a(n-12) -56*a(n-13) -70*a(n-16) +70*a(n-17) +56*a(n-20) -56*a(n-21) -28*a(n-24) +28*a(n-25) +8*a(n-28) -8*a(n-29) -a(n-32) +a(n-33) for n>43.

%F Empirical for n mod 4 = 0: a(n) = (1/2880)*n^8 + (1/35)*n^7 + (169/160)*n^6 + (91/12)*n^5 - (170437/960)*n^4 - (4291/60)*n^3 + (547067/45)*n^2 - (1847017/42)*n - 22525 for n>10.

%F Empirical for n mod 4 = 1: a(n) = (1/2880)*n^8 + (1/35)*n^7 + (1493/1440)*n^6 + (871/120)*n^5 - (506711/2880)*n^4 + (3067/30)*n^3 + (1371401/120)*n^2 - (11084513/210)*n + (371127/16) for n>10.

%F Empirical for n mod 4 = 2: a(n) = (1/2880)*n^8 + (13/504)*n^7 + (85/96)*n^6 + (2389/720)*n^5 - (50539/320)*n^4 + (93863/144)*n^3 + (142985/18)*n^2 - (26157409/420)*n + (399767/4) for n>10.

%F Empirical for n mod 4 = 3: a(n) = (1/2880)*n^8 + (79/2520)*n^7 + (187/160)*n^6 + (7409/720)*n^5 - (190777/960)*n^4 - (235699/720)*n^3 + (5757523/360)*n^2 - (52286503/840)*n + (146121/16) for n>10

%e Some solutions for n=4:

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

%e ..3..2..1....2..4..1....2..3..1....1..2..1....1..2..0....2..2..1....1..2..0

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

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

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

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

%e Knight distance matrix for n=4:

%e ..0..3..2

%e ..3..4..1

%e ..2..1..4

%e ..3..2..3

%e ..2..3..2

%e ..3..4..3

%K nonn

%O 1,1

%A _R. H. Hardin_, Dec 27 2014

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Last modified February 4 19:33 EST 2023. Contains 360059 sequences. (Running on oeis4.)