A257062 T(n,k)=Number of length n 1..(k+1) arrays with every leading partial sum divisible by 2 or 3.

1, 2, 2, 3, 4, 2, 3, 7, 6, 2, 4, 9, 18, 11, 3, 4, 16, 27, 45, 20, 4, 5, 18, 64, 81, 113, 33, 4, 6, 27, 81, 256, 243, 284, 59, 5, 7, 35, 141, 364, 1024, 729, 713, 104, 7, 7, 45, 200, 738, 1636, 4096, 2187, 1791, 178, 8, 8, 49, 293, 1149, 3866, 7353, 16384, 6561, 4498, 314, 9

Offset: 1

Author

R. H. Hardin, Apr 15 2015

Keywords

• nonn
• tabl

Comments

Table starts

.1...2.....3.....3.......4.......4........5........6.........7.........7

.2...4.....7.....9......16......18.......27.......35........45........49

.2...6....18....27......64......81......141......200.......293.......343

.2..11....45....81.....256.....364......738.....1149......1905......2401

.3..20...113...243....1024....1636.....3866.....6599.....12387.....16807

.4..33...284...729....4096....7353....20249....37893.....80545....117649

.4..59...713..2187...16384...33048...106056...217603....523733....823543

.5.104..1791..6561...65536..148534...555483..1249592...3405505...5764801

.7.178..4498.19683..262144..667585..2909419..7175812..22143847..40353607

.8.314.11297.59049.1048576.3000456.15238479.41207296.143987445.282475249

Examples

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

Links

• R. H. Hardin, Table of n, a(n) for n = 1..9999

Crossrefs

Column 1 is A079398(n+4)

Column 2 is A026385(n+1)

Column 4 is A000244

Column 5 is A000302

Formula

Empirical for column k:

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

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

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

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

k=5: a(n) = 4*a(n-1)

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

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

k=8: a(n) = 4*a(n-1) +8*a(n-2) +11*a(n-3) +3*a(n-4)

k=9: a(n) = 5*a(n-1) +9*a(n-2) +5*a(n-3)

k=10: a(n) = 7*a(n-1)

k=11: a(n) = 8*a(n-1)

k=12: a(n) = 8*a(n-1) +4*a(n-2) +2*a(n-3)

k=13: a(n) = 8*a(n-1) +10*a(n-2) +13*a(n-3) +7*a(n-4)

k=14: a(n) = 8*a(n-1) +15*a(n-2) +19*a(n-3) +5*a(n-4)

k=15: a(n) = 9*a(n-1) +15*a(n-2) +8*a(n-3)

k=16: a(n) = 11*a(n-1)

k=17: a(n) = 12*a(n-1)

k=18: a(n) = 12*a(n-1) +6*a(n-2) +3*a(n-3)

k=19: a(n) = 12*a(n-1) +15*a(n-2) +19*a(n-3) +10*a(n-4)

k=20: a(n) = 12*a(n-1) +22*a(n-2) +27*a(n-3) +7*a(n-4)

k=21: a(n) = 13*a(n-1) +21*a(n-2) +11*a(n-3)

k=22: a(n) = 15*a(n-1)

k=23: a(n) = 16*a(n-1)

Empirical for row n:

n=1: a(n) = a(n-1) +a(n-6) -a(n-7)

n=2: a(n) = a(n-1) +2*a(n-6) -2*a(n-7) -a(n-12) +a(n-13)

n=3: a(n) = a(n-1) +3*a(n-6) -3*a(n-7) -3*a(n-12) +3*a(n-13) +a(n-18) -a(n-19)

n=4: [order 25]

n=5: [order 29]

n=6: [order 37]

n=7: [order 43]

Empirical quasipolynomials for row n:

n=1: polynomial of degree 1 plus a quasipolynomial of degree 0 with period 6

n=2: polynomial of degree 2 plus a quasipolynomial of degree 1 with period 6

n=3: polynomial of degree 3 plus a quasipolynomial of degree 2 with period 6

n=4: polynomial of degree 4 plus a quasipolynomial of degree 3 with period 6

n=5: polynomial of degree 5 plus a quasipolynomial of degree 4 with period 6

n=6: polynomial of degree 6 plus a quasipolynomial of degree 5 with period 6

n=7: polynomial of degree 7 plus a quasipolynomial of degree 6 with period 6