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

Author

R. H. Hardin Feb 01 2013

Keywords

• nonn
• tabl

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

Examples

			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
		

Links

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

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

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