A223305 -id:A223305 - cp's OEIS frontend

A223299 4 X 4 X 4 triangular graph coloring a rectangular array: number of n X 2 0..9 arrays where 0..9 label nodes of a graph with edges 0,1 0,2 1,2 1,3 1,4 2,4 3,4 2,5 4,5 3,6 3,7 4,7 6,7 4,8 5,8 7,8 5,9 8,9 and every array movement to a horizontal or vertical neighbor moves along an edge of this graph.

36, 324, 3132, 30564, 298620, 2918052, 28515132, 278649828, 2722966524, 26608833828, 260021573820, 2540931306084, 24829985481084, 242638664618916, 2371065485035068, 23170056359958756, 226417834139125500

Offset: 1

R. H. Hardin, Mar 19 2013

nonn

Column 2 of A223305.

			Some solutions for n=3:
..0..2....2..0....8..9....5..4....7..8....0..2....4..8....3..4....3..4....1..3
..2..5....4..1....4..5....9..8....4..7....1..0....2..5....4..2....1..2....0..1
..5..2....1..2....1..4....5..9....7..3....0..2....1..2....8..4....2..1....1..3

R. H. Hardin, Table of n, a(n) for n = 1..210

Cf. A223305.

Empirical: a(n) = 11*a(n-1) - 12*a(n-2).
Conjectures from Colin Barker, Aug 19 2018: (Start)
G.f.: 36*x*(1 - 2*x) / (1 - 11*x + 12*x^2).
a(n) = (3*2^(-n)*((11-sqrt(73))^n*(-1+sqrt(73)) + (1+sqrt(73))*(11+sqrt(73))^n)) / sqrt(73).
(End)

A223300 4X4X4 triangular graph coloring a rectangular array: number of nX3 0..9 arrays where 0..9 label nodes of a graph with edges 0,1 0,2 1,2 1,3 1,4 2,4 3,4 2,5 4,5 3,6 3,7 4,7 6,7 4,8 5,8 7,8 5,9 8,9 and every array movement to a horizontal or vertical neighbor moves along an edge of this graph.

Original entry on oeis.org

144, 3132, 76788, 1882500, 46589268, 1148040516, 28402185108, 700525554180, 17321067229140, 427424478219972, 10564355081565972, 260773436696057988, 6443737539288281940, 159091220790299549508, 3930517719610405689492

Offset: 1

R. H. Hardin Mar 19 2013

nonn

Column 3 of A223305

			Some solutions for n=3
..2..1..4....7..4..1....7..4..2....1..2..1....9..8..5....9..8..4....5..4..3
..4..3..7....4..1..4....6..3..1....2..1..0....5..9..8....8..4..8....4..3..4
..7..4..3....1..4..8....7..4..2....1..2..1....9..5..9....9..8..7....8..4..7

R. H. Hardin, Table of n, a(n) for n = 1..210

Empirical: a(n) = 10*a(n-1) +500*a(n-2) -2878*a(n-3) -15863*a(n-4) +79004*a(n-5) +144196*a(n-6) -616760*a(n-7) -363504*a(n-8) +1348864*a(n-9) +396544*a(n-10) -600576*a(n-11) for n>12

A223301 4X4X4 triangular graph coloring a rectangular array: number of nX4 0..9 arrays where 0..9 label nodes of a graph with edges 0,1 0,2 1,2 1,3 1,4 2,4 3,4 2,5 4,5 3,6 3,7 4,7 6,7 4,8 5,8 7,8 5,9 8,9 and every array movement to a horizontal or vertical neighbor moves along an edge of this graph.

Original entry on oeis.org

576, 30564, 1882500, 118001928, 7442199324, 470062209228, 29706324866280, 1877636655597516, 118686588470619204, 7502399551962981216, 474244299620974794852, 29978170786284871362588, 1894997177822538169880160

Offset: 1

R. H. Hardin Mar 19 2013

nonn

Column 4 of A223305

			Some solutions for n=3
..1..4..3..7....0..2..5..9....0..1..4..8....1..3..1..4....5..2..5..2
..0..1..4..8....2..5..4..8....2..4..2..5....0..1..4..8....2..1..4..1
..1..3..7..4....5..4..8..9....4..7..4..2....2..4..8..7....4..3..1..4

R. H. Hardin, Table of n, a(n) for n = 1..210

Empirical: a(n) = 94*a(n-1) -1666*a(n-2) -33696*a(n-3) +1100794*a(n-4) -4373330*a(n-5) -96641317*a(n-6) +770139565*a(n-7) +2078815868*a(n-8) -30815999776*a(n-9) +9439379121*a(n-10) +508827706118*a(n-11) -740346550259*a(n-12) -3820163432470*a(n-13) +8313303065134*a(n-14) +11790417886127*a(n-15) -37600871363022*a(n-16) -5093748767428*a(n-17) +69398396589688*a(n-18) -28648799032000*a(n-19) -44260499027840*a(n-20) +28257350216960*a(n-21) +9517866926336*a(n-22) -7538371685376*a(n-23) -621543923712*a(n-24) +546106392576*a(n-25) for n>26

A223302 4X4X4 triangular graph coloring a rectangular array: number of nX5 0..9 arrays where 0..9 label nodes of a graph with edges 0,1 0,2 1,2 1,3 1,4 2,4 3,4 2,5 4,5 3,6 3,7 4,7 6,7 4,8 5,8 7,8 5,9 8,9 and every array movement to a horizontal or vertical neighbor moves along an edge of this graph.

Original entry on oeis.org

2304, 298620, 46589268, 7442199324, 1208825525124, 195984333263964, 31982581839418764, 5198236290762699924, 848677490133585108996, 138037800797986968319404, 22532346060071398807282764

Offset: 1

R. H. Hardin Mar 19 2013

nonn

Column 5 of A223305

			Some solutions for n=3
..0..1..2..1..4....0..1..3..7..8....0..1..4..2..4....0..1..2..4..7
..1..2..5..4..2....1..2..1..4..5....1..0..2..4..2....1..2..4..3..4
..0..1..2..1..4....2..0..2..5..9....2..1..4..1..4....0..1..3..4..3

R. H. Hardin, Table of n, a(n) for n = 1..210

A223303 4X4X4 triangular graph coloring a rectangular array: number of nX6 0..9 arrays where 0..9 label nodes of a graph with edges 0,1 0,2 1,2 1,3 1,4 2,4 3,4 2,5 4,5 3,6 3,7 4,7 6,7 4,8 5,8 7,8 5,9 8,9 and every array movement to a horizontal or vertical neighbor moves along an edge of this graph.

Original entry on oeis.org

9216, 2918052, 1148040516, 470062209228, 195984333263964, 82235766550254744, 34624128327772135980, 14597586270725996512680, 6158981902609383720107172, 2599381915639470430301409024, 1097252078673262192056218956776

Offset: 1

R. H. Hardin Mar 19 2013

nonn

Column 6 of A223305

			Some solutions for n=3
..0..1..0..1..4..8....0..1..0..2..4..7....0..1..4..1..3..1....0..1..4..1..3..1
..1..0..1..0..1..4....1..0..1..4..1..3....1..0..1..2..1..0....1..0..1..2..1..3
..0..1..2..1..4..2....0..1..3..7..3..1....0..1..2..1..3..1....0..1..0..1..4..1

R. H. Hardin, Table of n, a(n) for n = 1..210

A223304 4X4X4 triangular graph coloring a rectangular array: number of nX7 0..9 arrays where 0..9 label nodes of a graph with edges 0,1 0,2 1,2 1,3 1,4 2,4 3,4 2,5 4,5 3,6 3,7 4,7 6,7 4,8 5,8 7,8 5,9 8,9 and every array movement to a horizontal or vertical neighbor moves along an edge of this graph.

Original entry on oeis.org

36864, 28515132, 28402185108, 29706324866280, 31982581839418764, 34624128327772135980, 37878775658906097327552, 41350061045665728593745804, 45417884615768221932462708276, 49696230571397332338931201937088

Offset: 1

R. H. Hardin Mar 19 2013

nonn

Column 7 of A223305

			Some solutions for n=3
..0..1..0..2..1..2..4....0..1..0..2..4..8..4....0..1..0..2..4..7..4
..1..0..1..4..3..4..3....1..0..1..4..2..4..1....1..0..1..0..2..4..5
..0..1..0..2..1..2..1....0..1..0..1..4..1..3....0..1..0..1..0..2..4

R. H. Hardin, Table of n, a(n) for n = 1..210

A223298 4X4X4 triangular graph coloring a rectangular array: number of n X n 0..9 arrays where 0..9 label nodes of a graph with edges 0,1 0,2 1,2 1,3 1,4 2,4 3,4 2,5 4,5 3,6 3,7 4,7 6,7 4,8 5,8 7,8 5,9 8,9 and every array movement to a horizontal or vertical neighbor moves along an edge of this graph.

Original entry on oeis.org

10, 324, 76788, 118001928, 1208825525124, 82235766550254744, 37878775658906097327552, 117618648410211422628657954648, 2498412967821017756764735962549014076

Offset: 1

R. H. Hardin Mar 19 2013

nonn

Diagonal of A223305

			Some solutions for n=3
..2..5..8....1..2..4....4..1..3....7..8..4....4..7..3....8..7..6....8..4..8
..0..2..4....2..4..7....1..2..4....3..4..1....5..8..4....7..6..7....4..2..4
..2..0..1....4..1..4....0..1..2....4..2..4....9..5..2....4..3..6....5..4..1

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