A221967 T(n,k)=Number of -k..k arrays of length n with the sum ahead of each element differing from the sum following that element by k or less

3, 5, 9, 7, 25, 15, 9, 49, 65, 33, 11, 81, 175, 225, 63, 13, 121, 369, 833, 705, 129, 15, 169, 671, 2241, 3647, 2305, 255, 17, 225, 1105, 4961, 12609, 16513, 7425, 513, 19, 289, 1695, 9633, 34111, 73089, 73983, 24065, 1023, 21, 361, 2465, 17025, 78273, 241153

Offset

1,1

Comments

Table starts

....3.......5.........7..........9..........11...........13............15

....9......25........49.........81.........121..........169...........225

...15......65.......175........369.........671.........1105..........1695

...33.....225.......833.......2241........4961.........9633.........17025

...63.....705......3647......12609.......34111........78273........159615

..129....2305.....16513......73089......241153.......653185.......1535745

..255....7425.....73983.....419841.....1690623......5407233......14661375

..513...24065....332801....2419713....11888129.....44890625.....140355585

.1023...77825...1495039...13930497....83512319....372332545....1342437375

.2049..251905...6719489...80230401...586864641...3089205249...12843782145

.4095..815105..30195711..462012417..4123582463..25628045313..122870296575

.8193.2637825.135700481.2660655105.28975366145.212618141697.1175482548225

Links

R. H. Hardin, Table of n, a(n) for n = 1..334

Formula

Empirical for column k:

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

k=2: a(n) = 3*a(n-1) +2*a(n-2) -4*a(n-3)

k=3: a(n) = 3*a(n-1) +8*a(n-2) -4*a(n-3) -8*a(n-4)

k=4: a(n) = 5*a(n-1) +8*a(n-2) -20*a(n-3) -8*a(n-4) +16*a(n-5)

k=5: a(n) = 5*a(n-1) +18*a(n-2) -20*a(n-3) -48*a(n-4) +16*a(n-5) +32*a(n-6)

k=6: a(n) = 7*a(n-1) +18*a(n-2) -56*a(n-3) -48*a(n-4) +112*a(n-5) +32*a(n-6) -64*a(n-7)

k=7: a(n) = 7*a(n-1) +32*a(n-2) -56*a(n-3) -160*a(n-4) +112*a(n-5) +256*a(n-6) -64*a(n-7) -128*a(n-8)

Empirical for row n:

n=1: a(n) = 2*n + 1

n=2: a(n) = 4*n^2 + 4*n + 1

n=3: a(n) = 4*n^3 + 6*n^2 + 4*n + 1

n=4: a(n) = (16/3)*n^4 + (32/3)*n^3 + (32/3)*n^2 + (16/3)*n + 1

n=5: a(n) = (20/3)*n^5 + (50/3)*n^4 + 20*n^3 + (40/3)*n^2 + (16/3)*n + 1

n=6: a(n) = (128/15)*n^6 + (128/5)*n^5 + (112/3)*n^4 + 32*n^3 + (272/15)*n^2 + (32/5)*n + 1

n=7: a(n) = (488/45)*n^7 + (1708/45)*n^6 + (2912/45)*n^5 + (602/9)*n^4 + (2072/45)*n^3 + (952/45)*n^2 + (32/5)*n + 1

Example

Some solutions for n=6 k=4
..4...-2....4....1...-4...-1...-2....1...-2...-1....1....3....4....1...-1...-1
.-4....4...-4....0....4....4....3....2....3....2...-2...-4...-2...-3....3....3
..1...-3....3...-2...-1...-2...-3...-2...-2....2....0....3....1....2....0...-1
..0....2...-1....3...-2....0....2....2....3...-3....4....1...-3...-2...-2....1
..3...-4...-2...-3....3....3...-2....1...-1....0...-1...-3....0....3...-3...-2
..1....1....2....1...-1...-2...-1....1...-1....1....0....1....2...-4....4....2

Crossrefs

Column 1 is A062510(n+1)

Column 2 is A189318

Row 2 is A016754

Row 3 is A005917(n+1)

Row 4 is A142993

Sequence in context: A166722 A094549 A029642 * A079428 A094548 A112661

Adjacent sequences: A221964 A221965 A221966 * A221968 A221969 A221970

Keyword

nonn,tabl

Author

R. H. Hardin Feb 01 2013

Status

approved