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%I #7 Dec 27 2014 18:21:15
%S 14541,139760,2801464,28256241,343085240,3518586637,32095583880,
%T 235545304374,2365327348752,6157835478824,61076262847690,
%U 134095009274344,607685624163428,879408920721914,4881736615145034,7415366603022735
%N Number of (n+2)X(5+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 5 of A253119.
%H R. H. Hardin, <a href="/A253116/b253116.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>63.
%F Empirical for n mod 4 = 0: a(n) = (153041764352/315)*n^8 - (15010470559744/315)*n^7 + (6961612185856/3)*n^6 - (3236674053410617/45)*n^5 + (46748474263982980789/30720)*n^4 - (254905419128359866013/11520)*n^3 + (1718696267839192688635/8064)*n^2 - (2068610484101337801647/1680)*n + 3243081264128747149 for n>30.
%F Empirical for n mod 4 = 1: a(n) = (153041764352/315)*n^8 - (14551345266688/315)*n^7 + (99148717043456/45)*n^6 - (3032112744945337/45)*n^5 + (130257419296238992927/92160)*n^4 - (471054198709665535697/23040)*n^3 + (7040644141896011941503/35840)*n^2 - (20337258274642287345203/17920)*n + (6135450406615407750835/2048) for n>30.
%F Empirical for n mod 4 = 2: a(n) = (153041764352/315)*n^8 - (2253668483072/45)*n^7 + (114699616333568/45)*n^6 - (3699410997590201/45)*n^5 + (166373725910272083487/92160)*n^4 - (156762896891541573829/5760)*n^3 + (7304841953503218573653/26880)*n^2 - (781517497009749528281/480)*n + (569576081024807171947/128) for n>30.
%F Empirical for n mod 4 = 3: a(n) = (153041764352/315)*n^8 - (4595378814976/105)*n^7 + (17752656945920/9)*n^6 - (854801630750483/15)*n^5 + (104048190142177924127/92160)*n^4 - (118311522270222781609/7680)*n^3 + (8987239270067069299891/64512)*n^2 - (40603156976969041486523/53760)*n + (3819462220681350509763/2048) for n>30.
%e Some solutions for n=2:
%e ..0..2..2..3..2..2..3....0..1..1..1..1..1..2....0..2..1..1..0..2..2
%e ..2..3..1..2..2..3..2....1..2..1..1..2..2..2....1..2..0..1..1..2..1
%e ..2..1..3..2..2..2..2....1..0..2..2..1..2..2....1..0..2..1..1..1..2
%e ..3..1..2..1..2..3..2....3..1..1..1..2..2..1....3..0..1..1..2..2..1
%e Knight distance matrix for n=2:
%e ..0..3..2..3..2..3..4
%e ..3..4..1..2..3..4..3
%e ..2..1..4..3..2..3..4
%e ..5..2..3..2..3..4..3
%K nonn
%O 1,1
%A _R. H. Hardin_, Dec 27 2014
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