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A253114 Number of (n+2)X(3+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 1342,6334,64228,426571,2801464,11529235,79249122,269081698,

%T 1126140254,2033916546,10398959008,19253327106,49068963720,

%U 62663185107,216434297394,306964534797,600045409295,683103473164,1832436683833,2296747272574

%N Number of (n+2)X(3+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 3 of A253119

%H R. H. Hardin, <a href="/A253114/b253114.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>48.

%F Empirical for n mod 4 = 0: a(n) = (48128/315)*n^8 - (70784/45)*n^7 + (1613986/45)*n^6 - (408827287/360)*n^5 + (656806290529/46080)*n^4 - (417508867243/5760)*n^3 + (1340641181597/20160)*n^2 + (68326310189/120)*n - 1314231925 for n>15.

%F Empirical for n mod 4 = 1: a(n) = (48128/315)*n^8 - (399232/315)*n^7 + (494198/15)*n^6 - (391895399/360)*n^5 + (200444821003/15360)*n^4 - (719258439109/11520)*n^3 + (8134684965509/161280)*n^2 + (12687388527967/26880)*n - (1104560946555/1024) for n>15.

%F Empirical for n mod 4 = 2: a(n) = (48128/315)*n^8 - (784256/315)*n^7 + (254546/5)*n^6 - (494044999/360)*n^5 + (99930960601/5120)*n^4 - (787756612561/5760)*n^3 + (17328260184551/40320)*n^2 - (780223082251/3360)*n - (71148317231/64) for n>15.

%F Empirical for n mod 4 = 3: a(n) = (48128/315)*n^8 - (110464/315)*n^7 + (516898/45)*n^6 - (54346123/72)*n^5 + (321821523169/46080)*n^4 + (153162748967/11520)*n^3 - (67343538926807/161280)*n^2 + (8549722002355/5376)*n - (1254021550131/1024) for n>15.

%e Some solutions for n=3:

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

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

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

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

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

%e Knight distance matrix for n=3:

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

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

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

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

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

%K nonn

%O 1,1

%A _R. H. Hardin_, Dec 27 2014

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Last modified January 27 05:08 EST 2023. Contains 359836 sequences. (Running on oeis4.)