A252820 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

1, 2, 2, 4, 6, 4, 7, 17, 17, 7, 11, 40, 63, 40, 11, 16, 81, 187, 187, 81, 16, 22, 147, 468, 684, 468, 147, 22, 29, 246, 1032, 2078, 2078, 1032, 246, 29, 37, 387, 2067, 5490, 7564, 5490, 2067, 387, 37, 46, 580, 3840, 13015, 23664, 23664, 13015, 3840, 580, 46, 56, 836, 6716

Offset

1,2

Comments

Table starts

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

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

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

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

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

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

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

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

.37.580..6716..57338.385736.2147638.10248688..43017980.161986236..555724696

.46.836.11179.109549.839799.5289321.28359679.132916561.555724696.2106102800

Links

R. H. Hardin, Table of n, a(n) for n = 1..1680

Formula

Empirical for column k:

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

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

k=3: [polynomial of degree 6]

k=4: [polynomial of degree 8]

k=5: [polynomial of degree 10]

k=6: [polynomial of degree 12]

k=7: [polynomial of degree 14]

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

Example

Some solutions for n=4 k=4
..0..1..2..2....0..1..2..3....0..0..1..1....0..1..2..3....0..0..0..1
..1..1..2..3....1..1..2..3....0..1..2..2....1..2..3..3....0..1..1..2
..2..2..3..4....1..1..2..3....1..2..3..3....2..3..3..4....1..2..2..3
..2..3..4..5....1..2..3..4....1..2..3..4....2..3..4..4....2..2..3..4

Crossrefs

Column 1 is A000124(n-1)

Sequence in context: A232065 A219741 A210603 * A218765 A219310 A220070

Adjacent sequences: A252817 A252818 A252819 * A252821 A252822 A252823

Keyword

nonn,tabl

Author

R. H. Hardin, Dec 22 2014

Status

approved