A257062 - OEIS

%I #4 Apr 15 2015 10:42:14

%S 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,

%T 6,27,81,256,243,284,59,5,7,35,141,364,1024,729,713,104,7,7,45,200,

%U 738,1636,4096,2187,1791,178,8,8,49,293,1149,3866,7353,16384,6561,4498,314,9

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

%C Table starts

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

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

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

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

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

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

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

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

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

%C .8.314.11297.59049.1048576.3000456.15238479.41207296.143987445.282475249

%H R. H. Hardin, <a href="/A257062/b257062.txt">Table of n, a(n) for n = 1..9999</a>

%F Empirical for column k:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

%F Empirical for row n:

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

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

%F 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)

%F n=4: [order 25]

%F n=5: [order 29]

%F n=6: [order 37]

%F n=7: [order 43]

%F Empirical quasipolynomials for row n:

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

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

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

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

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

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

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

%e Some solutions for n=4 k=4

%e ..2....2....2....4....4....4....4....2....3....2....3....3....3....4....4....3

%e ..4....2....4....5....2....5....2....2....3....2....5....5....3....4....2....3

%e ..4....4....4....1....3....5....2....2....4....5....1....2....2....2....2....3

%e ..2....1....4....2....5....4....1....4....2....5....5....4....1....5....4....5

%Y Column 1 is A079398(n+4)

%Y Column 2 is A026385(n+1)

%Y Column 4 is A000244

%Y Column 5 is A000302

%K nonn,tabl

%O 1,2

%A _R. H. Hardin_, Apr 15 2015