A223480 - OEIS

%I #4 Mar 20 2013 11:54:25

%S 1,3,20,9,27,400,27,135,243,8000,81,675,2025,2187,160000,243,3375,

%T 16875,30375,19683,3200000,729,16875,147825,421875,455625,177147,

%U 64000000,2187,84375,1296675,6526575,10546875,6834375,1594323,1280000000,6561,421875

%N T(n,k)=Rolling icosahedron face footprints: number of nXk 0..19 arrays starting with 0 where 0..19 label faces of an icosahedron and every array movement to a horizontal or antidiagonal neighbor moves across an icosahedral edge

%C Table starts

%C ..........1........3..........9...........27.............81..............243

%C .........20.......27........135..........675...........3375............16875

%C ........400......243.......2025........16875.........147825..........1296675

%C .......8000.....2187......30375.......421875........6526575........101331675

%C .....160000....19683.....455625.....10546875......288507825.......7939566675

%C ....3200000...177147....6834375....263671875....12755926575.....622332801675

%C ...64000000..1594323..102515625...6591796875...563999907825...48783753036675

%C .1280000000.14348907.1537734375.164794921875.24937217326575.3824122400271675

%H R. H. Hardin, <a href="/A223480/b223480.txt">Table of n, a(n) for n = 1..160</a>

%F Empirical for column k:

%F k=1: a(n) = 20*a(n-1)

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

%F k=3: a(n) = 15*a(n-1)

%F k=4: a(n) = 25*a(n-1)

%F k=5: a(n) = 51*a(n-1) -300*a(n-2)

%F k=6: a(n) = 101*a(n-1) -1900*a(n-2) +10000*a(n-3)

%F k=7: a(n) = 227*a(n-1) -14764*a(n-2) +411840*a(n-3) -5347200*a(n-4) +29600000*a(n-5) -48000000*a(n-6)

%F Empirical for row n:

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

%F n=2: a(n) = 5*a(n-1) for n>2

%F n=3: a(n) = 9*a(n-1) -2*a(n-2) for n>4

%F n=4: a(n) = 17*a(n-1) -16*a(n-2) -76*a(n-3) +64*a(n-4) for n>7

%F n=5: a(n) = 33*a(n-1) -86*a(n-2) -1564*a(n-3) +7040*a(n-4) -6480*a(n-5) -5088*a(n-6) +5824*a(n-7) -512*a(n-8) for n>13

%F n=6: [order 20] for n>25

%e Some solutions for n=3 k=4

%e ..0..1..6.10....0..1..4..1....0..1..0..1....0..2..8.13....0..2..8..9

%e ..6..1..6..1....6..1..4..3....4..1..0..5....0..2..8..2....8..9..8..9

%e ..4..1..4..3....4.17..4.17....0..5..9..5....8..2..3..4....8..2..8..9

%e Face neighbors:

%e 0 -> 1 2 5

%e 1 -> 0 4 6

%e 2 -> 0 3 8

%e 3 -> 2 4 16

%e 4 -> 3 1 17

%e 5 -> 0 7 9

%e 6 -> 1 7 10

%e 7 -> 6 5 11

%e 8 -> 2 9 13

%e 9 -> 8 5 14

%e 10 -> 6 12 17

%e 11 -> 7 12 14

%e 12 -> 11 10 19

%e 13 -> 8 15 16

%e 14 -> 9 11 15

%e 15 -> 14 13 19

%e 16 -> 3 13 18

%e 17 -> 4 10 18

%e 18 -> 16 17 19

%e 19 -> 15 18 12

%Y Column 1 is A009964(n-1)

%Y Column 2 is A013708(n-1)

%Y Column 3 is 9*15^(n-1)

%Y Column 4 is 27*25^(n-1)

%Y Row 1 is A000244(n-1)

%Y Row 2 is 27*5^(n-2) for n>1

%K nonn,tabl

%O 1,2

%A _R. H. Hardin_ Mar 20 2013