cp's OEIS Frontend

A223357 T(n,k)=Rolling cube face footprints: number of nXk 0..5 arrays starting with 0 where 0..5 label faces of a cube and every array movement to a horizontal or antidiagonal neighbor moves across a corresponding cube edge.

%I A223357 #6 Jul 23 2025 04:02:46
%S A223357 1,4,6,16,64,36,64,768,1024,216,256,9216,36864,16384,1296,1024,110592,
%T A223357 1327104,1769472,262144,7776,4096,1327104,48365568,191102976,84934656,
%U A223357 4194304,46656,16384,15925248,1764753408,21177040896,27518828544
%N A223357 T(n,k)=Rolling cube face footprints: number of nXk 0..5 arrays starting with 0 where 0..5 label faces of a cube and every array movement to a horizontal or antidiagonal neighbor moves across a corresponding cube edge.
%C A223357 Table starts
%C A223357 ....1.......4.........16............64..............256................1024
%C A223357 ....6......64........768..........9216...........110592.............1327104
%C A223357 ...36....1024......36864.......1327104.........48365568..........1764753408
%C A223357 ..216...16384....1769472.....191102976......21177040896.......2356125106176
%C A223357 .1296..262144...84934656...27518828544....9273505480704....3147420753985536
%C A223357 .7776.4194304.4076863488.3962711310336.4060947412942848.4204783428144463872
%H A223357 R. H. Hardin, <a href="/A223357/b223357.txt">Table of n, a(n) for n = 1..241</a>
%F A223357 Empirical for column k:
%F A223357 k=1: a(n) = 6*a(n-1)
%F A223357 k=2: a(n) = 16*a(n-1)
%F A223357 k=3: a(n) = 48*a(n-1)
%F A223357 k=4: a(n) = 144*a(n-1)
%F A223357 k=5: a(n) = 480*a(n-1) -18432*a(n-2)
%F A223357 k=6: a(n) = 1600*a(n-1) -368640*a(n-2) +21233664*a(n-3)
%F A223357 k=7: a(n) = 5376*a(n-1) -5750784*a(n-2) +2038431744*a(n-3) -217432719360*a(n-4)
%F A223357 Empirical for row n:
%F A223357 n=1: a(n) = 4*a(n-1)
%F A223357 n=2: a(n) = 12*a(n-1) for n>2
%F A223357 n=3: a(n) = 40*a(n-1) -128*a(n-2) for n>4
%F A223357 n=4: a(n) = 144*a(n-1) -3840*a(n-2) +24576*a(n-3) for n>7
%F A223357 n=5: [order 7] for n>11
%F A223357 n=6: [order 9] for n>15
%F A223357 n=7: [order 27] for n>33
%e A223357 Some solutions for n=3 k=4
%e A223357 ..0..2..0..2....0..3..5..2....0..2..5..4....0..4..0..4....0..1..3..4
%e A223357 ..0..4..0..1....0..3..5..3....0..4..0..1....0..4..0..4....0..1..0..1
%e A223357 ..0..4..5..4....0..1..0..1....0..4..2..1....0..2..0..1....0..1..2..4
%Y A223357 Column 1 is A000400(n-1)
%Y A223357 Column 2 is A013709 (n-1)
%Y A223357 Column 3 is 16*48^(n-1)
%Y A223357 Column 4 is 64*144^(n-1)
%Y A223357 Row 1 is A000302(n-1)
%Y A223357 Row 2 is 64*12^(n-2) for n>1
%K A223357 nonn,tabl
%O A223357 1,2
%A A223357 _R. H. Hardin_ Mar 19 2013