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%I #9 Sep 27 2015 12:20:39
%S 87621,1472184,32653458,621408160,11880244815,198343663311,
%T 3788905574401,65148353195615,946743614051298,12716755528646645,
%U 234941253852401152,1713534280746677578,26771382269792112043,109318218432421573086
%N Number of (n+2)X(6+2) nonnegative integer arrays with all values the knight distance from the upper left minus as much as 3, with successive minimum path knight move differences either 0 or +1, and any unreachable value zero.
%C Column 6 of A253342.
%H R. H. Hardin, <a href="/A253340/b253340.txt">Table of n, a(n) for n = 1..171</a>
%F Empirical: a(n) = a(n-1) +12*a(n-2) -12*a(n-3) -66*a(n-4) +66*a(n-5) +220*a(n-6) -220*a(n-7) -495*a(n-8) +495*a(n-9) +792*a(n-10) -792*a(n-11) -924*a(n-12) +924*a(n-13) +792*a(n-14) -792*a(n-15) -495*a(n-16) +495*a(n-17) +220*a(n-18) -220*a(n-19) -66*a(n-20) +66*a(n-21) +12*a(n-22) -12*a(n-23) -a(n-24) +a(n-25) for n>93.
%F Empirical for n mod 2 = 0: a(n) = (41845195937712963584/467775)*n^12 - (787696630451469811712/31185)*n^11 + (148962346048411016364032/42525)*n^10 - (888428634222323516309504/2835)*n^9 + (283740371281858118888751104/14175)*n^8 - (180763383638626754347121536/189)*n^7 + (11867756870602406424419355452863/340200)*n^6 - (22105596431914088616726161901349/22680)*n^5 + (7018796941146059593727898272145527/340200)*n^4 - (912029795935048532055594220201528/2835)*n^3 + (1454714137102535633473169697829759199/415800)*n^2 - (329496280782634405010285105419432573/13860)*n + 76183870652927257367576438898581 for n>68.
%F Empirical for n mod 2 = 1: a(n) = (41845195937712963584/467775)*n^12 - (342827488046045200384/14175)*n^11 + (27430752363819500044288/8505)*n^10 - (263128784881075022200832/945)*n^9 + (243996148888268289577877504/14175)*n^8 - (3769923204527648981879181184/4725)*n^7 + (384925024924715015913656648599/13608)*n^6 - (17451185860321510562243619956219/22680)*n^5 + (5402173850133052765550507074283327/340200)*n^4 - (4568122631102080015587010798528177/18900)*n^3 + (106802994227879624350633195648348577/41580)*n^2 - (14341129594636450057761369204929629/840)*n + (107129466593833911479814437032315/2) for n>68.
%e Some solutions for n=1
%e ..0..2..1..2..1..1..2..2....0..1..1..1..1..2..3..3....0..3..2..3..2..2..3..3
%e ..1..2..1..1..1..2..2..2....1..2..0..1..2..2..2..2....2..3..1..2..2..3..2..3
%e ..1..1..2..1..1..1..1..3....0..0..2..2..1..2..2..3....2..1..3..3..2..2..3..4
%e Knight distance matrix for n=1
%e ..0..3..2..3..2..3..4..5
%e ..3..4..1..2..3..4..3..4
%e ..2..1..4..3..2..3..4..5
%K nonn
%O 1,1
%A _R. H. Hardin_, Dec 30 2014
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