A253335 - OEIS

%I #9 Sep 27 2015 12:21:07

%S 69,488,1928,7494,27015,87621,319172,945613,2874539,6935762,22983454,

%T 49969332,119548267,214145613,622017082,1069718995,2070309209,

%U 3028231177,7700984072,11401562937,19017598555,24702675207,56114922678,75683326471

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

%C Column 1 of A253342.

%H R. H. Hardin, <a href="/A253335/b253335.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = a(n-1) +12*a(n-4) -12*a(n-5) -66*a(n-8) +66*a(n-9) +220*a(n-12) -220*a(n-13) -495*a(n-16) +495*a(n-17) +792*a(n-20) -792*a(n-21) -924*a(n-24) +924*a(n-25) +792*a(n-28) -792*a(n-29) -495*a(n-32) +495*a(n-33) +220*a(n-36) -220*a(n-37) -66*a(n-40) +66*a(n-41) +12*a(n-44) -12*a(n-45) -a(n-48) +a(n-49) for n>66

%F Empirical for n mod 4 = 0: a(n) = (1/34214400)*n^12 + (19/3326400)*n^11 + (11413/21772800)*n^10 + (2543/120960)*n^9 + (171211/1036800)*n^8 - (17681/1800)*n^7 - (364775123/3110400)*n^6 + (503575579/120960)*n^5 - (23104979617/5443200)*n^4 - (42603595499/75600)*n^3 + (271867785329/59400)*n^2 - (1488535171/330)*n - 43423533 for n>17

%F Empirical for n mod 4 = 1: a(n) = (1/34214400)*n^12 + (19/3326400)*n^11 + (11329/21772800)*n^10 + (139/6720)*n^9 + (1113277/7257600)*n^8 - (80513/8400)*n^7 - (321868919/3110400)*n^6 + (24501551/6048)*n^5 - (2717152987/340200)*n^4 - (13345114753/25200)*n^3 + (32592081480029/6652800)*n^2 - (327571973557/36960)*n - (2047419033/64) for n>17

%F Empirical for n mod 4 = 2: a(n) = (1/34214400)*n^12 + (107/19958400)*n^11 + (1453/3110400)*n^10 + (12203/725760)*n^9 + (59611/1036800)*n^8 - (2635259/302400)*n^7 - (138251441/3110400)*n^6 + (2772874619/725760)*n^5 - (630113333/24300)*n^4 - (150767030627/453600)*n^3 + (1190430331397/237600)*n^2 - (972074050223/55440)*n - (25910239/4) for n>17

%F Empirical for n mod 4 = 3: a(n) = (1/34214400)*n^12 + (11/1814400)*n^11 + (12403/21772800)*n^10 + (3487/145152)*n^9 + (1620937/7257600)*n^8 - (1082489/100800)*n^7 - (453300833/3110400)*n^6 + (3176992123/725760)*n^5 + (6382948247/1360800)*n^4 - (646363851967/907200)*n^3 + (35282024010953/6652800)*n^2 - (80650610293/20160)*n - (3453848023/64) for n>17

%e Some solutions for n=4

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

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

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

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

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

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

%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 30 2014