%I #4 Dec 22 2014 10:52:57

%S 1,2,2,4,6,4,7,17,17,7,11,40,63,40,11,16,81,187,187,81,16,22,147,468,

%T 684,468,147,22,29,246,1032,2078,2078,1032,246,29,37,387,2067,5490,

%U 7564,5490,2067,387,37,46,580,3840,13015,23664,23664,13015,3840,580,46,56,836,6716

%N T(n,k)=Number of nXk nonnegative integer arrays with upper left 0 and every value within 2 of its city block distance from the upper left and every value increasing by 0 or 1 with every step right or down

%C Table starts

%C ..1...2.....4......7.....11......16.......22........29........37.........46

%C ..2...6....17.....40.....81.....147......246.......387.......580........836

%C ..4..17....63....187....468....1032.....2067......3840......6716......11179

%C ..7..40...187....684...2078....5490....13015.....28299.....57338.....109549

%C .11..81...468...2078...7564...23664....65711....165685....385736.....839799

%C .16.147..1032...5490..23664...86724...279300....809349...2147638....5289321

%C .22.246..2067..13015..65711..279300..1033761...3414257..10248688...28359679

%C .29.387..3840..28299.165685..809349..3414257..12755742..43017980..132916561

%C .37.580..6716..57338.385736.2147638.10248688..43017980.161986236..555724696

%C .46.836.11179.109549.839799.5289321.28359679.132916561.555724696.2106102800

%H R. H. Hardin, <a href="/A252820/b252820.txt">Table of n, a(n) for n = 1..1680</a>

%F Empirical for column k:

%F k=1: a(n) = (1/2)*n^2 - (1/2)*n + 1

%F k=2: a(n) = (1/24)*n^4 + (5/12)*n^3 - (1/24)*n^2 + (7/12)*n + 1

%F k=3: [polynomial of degree 6]

%F k=4: [polynomial of degree 8]

%F k=5: [polynomial of degree 10]

%F k=6: [polynomial of degree 12]

%F k=7: [polynomial of degree 14]

%F Empirical for "within 1" instead of "within 2" is T(n,k)=binomial(n+k,k)-1

%e Some solutions for n=4 k=4

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

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

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

%e ..2..3..4..5....1..2..3..4....1..2..3..4....2..3..4..4....2..2..3..4

%Y Column 1 is A000124(n-1)

%K nonn,tabl

%O 1,2

%A _R. H. Hardin_, Dec 22 2014