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
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
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
Comments