Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #7 Dec 27 2014 18:20:44
%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