%I #4 Dec 22 2014 11:11:04

%S 1,2,2,4,6,4,8,18,18,8,15,53,81,53,15,26,142,340,340,142,26,42,339,

%T 1238,1920,1238,339,42,64,729,3891,9075,9075,3891,729,64,93,1437,

%U 10761,36292,54376,36292,10761,1437,93,130,2638,26764,125892,271846,271846,125892

%N T(n,k)=Number of nXk nonnegative integer arrays with upper left 0 and every value within 3 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.......8.......15........26.........42..........64...........93

%C ...2....6.....18......53......142.......339........729........1437.........2638

%C ...4...18.....81.....340.....1238......3891......10761.......26764........60988

%C ...8...53....340....1920.....9075.....36292.....125892......387849......1082111

%C ..15..142...1238....9075....54376....271846....1165921.....4396009.....14863460

%C ..26..339...3891...36292...271846...1679072....8807722....40232545....163307844

%C ..42..729..10761..125892..1165921...8807722...55960651...306796310...1481748658

%C ..64.1437..26764..387849..4396009..40232545..306796310..2001017650..11403395172

%C ..93.2638..60988.1082111.14863460.163307844.1481748658.11403395172..76084625352

%C .130.4568.129236.2777103.45791493.598768118.6411737114.57777817522.448101581256

%H R. H. Hardin, <a href="/A252828/b252828.txt">Table of n, a(n) for n = 1..721</a>

%F Empirical for column k:

%F k=1: a(n) = (1/6)*n^3 - (1/2)*n^2 + (4/3)*n

%F k=2: [polynomial of degree 6]

%F k=3: [polynomial of degree 9]

%F k=4: [polynomial of degree 12]

%F k=5: [polynomial of degree 15]

%F k=6: [polynomial of degree 18]

%F k=7: [polynomial of degree 21]

%F Empirical for "within 1" instead of "within 3" is T(n,k)=binomial(n+k,k)-1

%e Some solutions for n=4 k=4

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

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

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

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

%Y Column 1 is A000125(n-1)

%K nonn,tabl

%O 1,2

%A _R. H. Hardin_, Dec 22 2014