A213089 -id:A213089 - cp's OEIS frontend

A213342 Irregular array T(n,k) of numbers/2 of non-extendable non-self-adjacent simple paths of each length within a square lattice bounded by rectangles with nodal dimensions n and 4, n >= 2.

4, 4, 6, 6, 4, 8, 16, 18, 14, 8, 14, 4, 8, 20, 36, 44, 24, 40, 16, 84, 4, 8, 20, 40, 72, 80, 90, 66, 184, 72, 236, 26, 4, 8, 20, 40, 78, 116, 192, 180, 354, 278, 530, 268, 546, 124, 32, 4, 8, 20, 40, 80, 122, 244, 336, 628, 628, 1130, 788, 1362, 878, 1168, 354, 292, 16

Offset: 2

Christopher Hunt Gribble, Jun 09 2012

The irregular array of numbers is:
...k..3....4....5....6....7....8....9...10...11...12...13...14...15...16...17...18...19...20
.n
.2....4....4....6....6
.3....4....8...16...18...14....8...14
.4....4....8...20...36...44...24...40...16...84
.5....4....8...20...40...72...80...90...66..184...72..236...26
.6....4....8...20...40...78..116..192..180..354..278..530..268..546..124...32
.7....4....8...20...40...80..122..244..336..628..628.1130..788.1362..878.1168..354..292...16
where k is the path length in nodes.
In an attempt to define the irregularity of the array, it appears that the maximum value of k is 3n for 2 <= n <= 3, 3n-1 for n = 4 and 3n - floor((n-2)/3) for n >= 5. Reading this array by rows gives the sequence. One half of the numbers of paths constitute the sequence to remove the effect of the bilateral symmetry of the rectangle.

			T(2,3) = One half of the number of complete non-self-adjacent simple paths of length 3 nodes within a square lattice bounded by a 2 X 4 node rectangle.

Cf. A213106, A213249, A213274, A213089.

A213375 Irregular array T(n,k) of numbers/2 of non-extendable (complete) non-self-adjacent simple paths of each length within a square lattice bounded by rectangles with nodal dimensions n and 5, n >= 2.

Original entry on oeis.org

4, 4, 6, 10, 10, 2, 4, 8, 16, 22, 42, 24, 42, 22, 18, 4, 8, 20, 40, 72, 80, 90, 66, 184, 72, 236, 26, 4, 8, 20, 44, 100, 136, 220, 156, 348, 244, 800, 336, 1308, 248, 56, 4, 8, 20, 44, 106, 172, 322, 410, 612, 602, 1462, 1122, 3240, 1712, 4682, 1394, 706, 218, 4

Offset: 2

Christopher Hunt Gribble, Jun 10 2012

The irregular array of numbers is:
...k..3....4....5....6....7....8....9...10...11...12...13...14...15...16....17...18....19...20...21...22...23...24
.n
.2....4....4....6...10...10....2
.3....4....8...16...22...42...24...42...22...18
.4....4....8...20...40...72...80...90...66..184...72..236...26
.5....4....8...20...44..100..136..220..156..348..244..800..336.1308..248....56
.6....4....8...20...44..106..172..322..410..612..602.1462.1122.3240.1712..4682.1394...706..218...4
where k is the path length in nodes. In an attempt to define the irregularity of the array, it appears that the maximum value of k is 3n+2 for 2 <= n <= 5, 3n+3 for 6 <= n <= 9 and 3n+4 for n >= 10. Reading this array by rows gives the sequence. One half of the numbers of paths constitute the sequence to remove the effect of the bilateral symmetry of the rectangle.

			T(2,3) = One half of the number of complete non-self-adjacent simple paths of length 3 nodes within a square lattice bounded by a 2 X 5 node rectangle.

Cf. A213106, A213249, A213274, A213089, A213342.

A213954 Irregular array T(n,k) of the numbers of non-extendable (complete) non-self-adjacent simple paths starting at each of a minimal subset of nodes within a square lattice bounded by rectangles with nodal dimensions n and 3, n >= 2.

Original entry on oeis.org

3, 4, 8, 6, 6, 8, 17, 14, 12, 10, 36, 32, 25, 18, 20, 12, 77, 68, 51, 36, 38, 20, 164, 142, 106, 72, 72, 38, 64, 28, 347, 298, 225, 146, 142, 74, 109, 46, 732, 628, 476, 302, 294, 148, 197, 82, 168, 64, 1543, 1324, 1003, 632, 614, 304, 385, 156, 277, 100, 3252, 2790, 2112, 1328, 1284, 634, 777, 312, 504, 174, 414, 136

Offset: 2

Christopher Hunt Gribble, Jun 30 2012

The subset of nodes approximately defines the top left-hand quarter of the rectangle and has nodal dimensions floor((n+1)/2) and 2 to capture all geometrically distinct counts.
The quarter-rectangle is read by rows.
The irregular array of numbers is:
....k.....1....2....3....4....5....6....7....8....9...10...11...12
..n
..2.......3....4
..3.......8....6....6....8
..4......17...14...12...10
..5......36...32...25...18...20...12
..6......77...68...51...36...38...20
..7.....164..142..106...72...72...38...64...28
..8.....347..298..225..146..142...74..109...46
..9.....732..628..476..302..294..148..197...82..168...64
.10....1543.1324.1003..632..614..304..385..156..277..100
.11....3252.2790.2112.1328.1284..634..777..312..504..174..414..136
where k indicates the position of the start node in the quarter-rectangle.
For each n, the maximum value of k is 2*floor((n+1)/2).
Reading this array by rows gives the sequence.

			When n = 2, the number of times (NT) each node in the rectangle is the start node (SN) of a complete non-self-adjacent simple path is
SN 0 1 2
   3 4 5
NT 3 4 3
   3 4 3
To limit duplication, only the top left-hand corner 3 and the 4 to its right are stored in the sequence, i.e. T(2,1) = 3 and T(2,2) = 4.

Cf. A213106, A213249, A213089, A213478.

It appears that:
T(n,1) - 2*T(n-1,1) - T(n-4,1) - 2 = 0, n >= 6
T(n,2) - 2*T(n-1,2) - T(n-4,1) = 0, n >= 6
T(n,3) - 2*T(n-1,3) - T(n-4,1) = 0, n >= 10
T(n,4) - 2*T(n-1,4) - T(n-4,1) + 8 = 0, n >= 7

A214121 Irregular array T(n,k) of the numbers of non-extendable (complete) non-self-adjacent simple paths ending at each of a minimal subset of nodes within a square lattice bounded by rectangles with nodal dimensions n and 3, n >= 2.

Original entry on oeis.org

5, 0, 14, 2, 2, 0, 33, 4, 6, 0, 75, 6, 13, 0, 16, 0, 165, 8, 27, 0, 32, 0, 353, 10, 57, 0, 62, 0, 60, 0, 747, 12, 119, 0, 124, 0, 109, 0, 1577, 14, 247, 0, 250, 0, 206, 0, 184, 0, 3327, 16, 515, 0, 508, 0, 399, 0, 323, 0, 7015, 18, 1079, 0, 1046, 0, 790, 0, 590

Offset: 2

Christopher Hunt Gribble, Jul 04 2012

The subset of nodes is contained in the top left-hand quarter of the rectangle and has nodal dimensions floor((n+1)/2) and 2 to capture all geometrically distinct counts. The quarter-rectangle is read by rows. The irregular array of numbers is:
....k.....1.....2.....3.....4.....5.....6.....7.....8.....9....10....11....12
..n
..2.......5.....0
..3......14.....2.....2.....0
..4......33.....4.....6.....0
..5......75.....6....13.....0....16.....0
..6.....165.....8....27.....0....32.....0
..7.....353....10....57.....0....62.....0....60.....0
..8.....747....12...119.....0...124.....0...109.....0
..9....1577....14...247.....0...250.....0...206.....0...184.....0
.10....3327....16...515.....0...508.....0...399.....0...323.....0
.11....7015....18..1079.....0..1046.....0...790.....0...590.....0...520.....0
.12...14785....20..2267.....0..2176.....0..1601.....0..1121.....0...877.....0
where k indicates the position of the end node in the quarter-rectangle. For each n, the maximum value of k is 2*floor((n+1)/2). Reading this array by rows gives the sequence.

			When n = 2, the number of times (NT) each node in the rectangle is the end node (EN) of a complete non-self-adjacent simple path is
EN 0 1 2
   3 4 5
NT 5 0 5
   5 0 5
To limit duplication, only the top left-hand corner 5 and the 0 to its right are stored in the sequence, i.e. T(2,1) = 5 and T(2,2) = 0.

Cf. A213106, A213249, A213089, A213954, A214119.

Let T(n,k) denote an element of the irregular array then it appears that
T(n,k) = 0, n >= 3, k = 2j, j >= 2,
T(n,1) - 2T(n-1,1) - T(n-4,1) - 8 = 0, n >= 8,
T(n,2) = 2(n-2), n >= 2,
T(n,3) - 2T(n-1,3) - T(n-4,3) + 2(n-7) = 0, n >= 9,
T(n,5) - 2T(n-1,5) - T(n-4,5) + 8(n-7) = 0, n >= 10,
T(n,7) - 2T(n-1,7) - T(n-4,7) + 20(n-8) + 8 = 0, n >= 11,
T(n,9) - 2T(n-1,9) - T(n-4,9) + 46(n-9) + 30 = 0, n >= 13,
T(n,11) - 2T(n-1,11) - T(n-4,11) + 104(n-10) + 84 = 0, n >= 15,
T(n,13) - 2T(n-1,13) - T(n-4,13) + 226(n-11) + 202 = 0, n >= 15.

Comment corrected by Christopher Hunt Gribble, Jul 22 2012

A213379 Irregular array T(n,k) of numbers/2 of non-extendable (complete) non-self-adjacent simple paths of each length within a square lattice bounded by rectangles with nodal dimensions n and 6, n >= 2.

Original entry on oeis.org

4, 4, 6, 10, 14, 16, 8, 4, 8, 16, 22, 48, 60, 82, 90, 66, 34, 24, 2, 4, 8, 20, 40, 78, 116, 192, 180, 354, 278, 530, 268, 546, 124, 32, 4, 8, 20, 44, 106, 172, 322, 410, 612, 602, 1462, 1122, 3240, 1712, 4682, 1394, 706, 218, 4

Offset: 2

Christopher Hunt Gribble, Jun 10 2012

The irregular array of numbers is:
...k..3....4....5....6....7....8....9...10...11...12...13...14...15...16...17...18...19...20...21
.n
.2....4....4....6...10...14...16....8
.3....4....8...16...22...48...60...82...90...66...34...24....2
.4....4....8...20...40...78..116..192..180..354..278..530..268..546..124...32
.5....4....8...20...44..106..172..322..410..612..602.1462.1122.3240.1712.4682.1394..706..218....4
where k is the path length in nodes. In an attempt to define the irregularity of the array, it appears that the maximum value of k is 4n - floor((n-8)/4) for n >= 8. Reading this array by rows gives the sequence. One half of the numbers of paths constitute the sequence to remove the effect of the bilateral symmetry of the rectangle.

			T(2,3) = One half of the number of complete non-self-adjacent simple paths of length 3 nodes within a square lattice bounded by a 2 X 6 node rectangle.

Cf. A213106, A213249, A213274, A213089, A213342, A213375.

A214504 Irregular array T(n,k) of the numbers of non-extendable (complete) non-self-adjacent simple paths incorporating each of a minimal subset of nodes within a square lattice bounded by rectangles with nodal dimensions n and 3, n >= 2.

Original entry on oeis.org

12, 14, 32, 36, 36, 48, 80, 88, 86, 100, 188, 210, 209, 228, 204, 204, 418, 470, 472, 524, 479, 452, 906, 1016, 1028, 1152, 1050, 1020, 1088, 980, 1943, 2170, 2219, 2472, 2250, 2222, 2333, 2200, 4137, 4610, 4754, 5260, 4811, 4738, 4929, 4784, 4920, 4924

Offset: 2

Christopher Hunt Gribble, Jul 19 2012

The subset of nodes is contained in the top left-hand quarter of the rectangle and has nodal dimensions floor((n+1)/2) and 2 to capture all geometrically distinct counts.
The quarter-rectangle is read by rows.
The irregular array of numbers is:
....k....1....2....3....4....5....6....7....8....9...10
..n
..2.....12...14
..3.....32...36...36...48
..4.....80...88...86..100
..5....188..210..209..228..204..204
..6....418..470..472..524..479..452
..7....906.1016.1028.1152.1050.1020.1088..980
..8...1943.2170.2219.2472.2250.2222.2333.2200
..9...4137.4610.4754.5260.4811.4738.4929.4784.4920.4924
where k indicates the position of a node in the quarter-rectangle.
For each n, the maximum value of k is 2*floor((n+1)/2).
Reading this array by rows gives the sequence.

			When n = 2, the number of times (NT) each node in the rectangle (N) occurs in a complete non-self-adjacent simple path is
N   0  1  2
    3  4  5
NT 12 14 12
   12 14 12
To limit duplication, only the top left-hand corner 12 and the 14 to its right are stored in the sequence,
  i.e. T(2,1) = 12 and T(2,2) = 14.

Cf. A213106, A213249, A213089, A213954, A214121, A214397, A214399

Comment corrected by Christopher Hunt Gribble, Jul 22 2012

A213383 Irregular array T(n,k) of numbers/2 of non-extendable (complete) non-self-adjacent simple paths of each length within a square lattice bounded by rectangles with nodal dimensions n and 7, n >= 2.

Original entry on oeis.org

4, 4, 6, 10, 14, 20, 26, 18, 2, 4, 8, 16, 22, 50, 66, 132, 160, 218, 120, 122, 56, 36, 4, 4, 8, 20, 40, 80, 122, 244, 336, 628, 628, 1130, 788, 1362, 878, 1168, 354, 292, 16

Offset: 2

Christopher Hunt Gribble, Jun 10 2012

The irregular array of numbers is:
...k..3....4....5....6....7....8....9...10...11...12...13...14...15...16...17...18...19...20
.n
.2....4....4....6...10...14...20...26...18....2
.3....4....8...16...22...50...66..132..160..218..120..122...56...36....4
.4....4....8...20...40...80..122..244..336..628..628.1130..788.1362..878.1168..354..292...16
where k is the path length in nodes. There is insufficient evidence to attempt to define the irregularity of the array.  However, the maximum values of k for 2 <= n <= 11 are 11, 16, 20, 24, 29, 33, 38, 42, 46, 50. Reading this array by rows gives the sequence. One half of the numbers of paths constitute the sequence to remove the effect of the bilateral symmetry of the rectangle.

			T(2,3) = One half of the number of complete non-self-adjacent simple paths of length 3 nodes within a square lattice bounded by a 2 X 7 node rectangle.

Cf. A213106, A213249, A213274, A213089, A213342, A213375, A213379.

A213425 Irregular array T(n,k) of numbers/2 of non-extendable (complete) non-self-adjacent simple paths of each length within a square lattice bounded by rectangles with nodal dimensions n and 8, n >= 2.

Original entry on oeis.org

4, 4, 6, 10, 14, 20, 30, 40, 34, 10, 4, 8, 16, 22, 52, 68, 144, 222, 334, 406, 302, 288, 198, 88, 52, 6, 4, 8, 20, 40, 82, 124, 258, 400, 894, 1098, 1984, 1960, 2796, 2388, 3426, 2290, 2638, 1008, 1316, 152

Offset: 2

Christopher Hunt Gribble, Jun 11 2012

The irregular array of numbers is:
...k..3....4....5....6....7....8....9...10...11...12...13...14...15...16...17...18...19...20...21...22
.n
.2....4....4....6...10...14...20...30...40...34...10
.3....4....8...16...22...52...68..144..222..334..406..302..288..198...88...52....6
.4....4....8...20...40...82..124..258..400..894.1098.1984.1960.2796.2388.3426.2290.2638.1008.1316..152
where k is the path length in nodes. There is insufficient evidence to attempt to define the irregularity of the array. However, the maximum values of k for 2 <= n <= 10 are 12, 18, 22, 27, 32, 38, 42, 48, 52. Reading this array by rows gives the sequence. One half of the numbers of paths constitute the sequence to remove the effect of the bilateral symmetry of the rectangle.

			T(2,3) = One half of the number of complete non-self-adjacent simple paths of length 3 nodes within a square lattice bounded by a 2 X 8 node rectangle.

Cf. A213106, A213249, A213274, A213089, A213342, A213375, A213379, A213383.

A213426 Irregular array T(n,k) of numbers/2 of non-extendable (complete) non-self-adjacent simple paths of each length within a square lattice bounded by rectangles with nodal dimensions n and 9, n >= 2.

Original entry on oeis.org

4, 4, 6, 10, 14, 20, 30, 44, 60, 60, 28, 2, 4, 8, 16, 22, 54, 70, 152, 238, 416, 574, 810, 642, 760, 456, 320, 136, 72, 8, 4, 8, 20, 40, 84, 126, 268, 418, 1014, 1450, 2890, 3510, 5474, 5286, 7238, 6926, 8218, 5636, 6754, 2956, 4220, 778, 48

Offset: 2

Christopher Hunt Gribble, Jun 11 2012

The irregular array of numbers is:
...k..3....4....5....6....7....8....9...10...11...12...13...14...15...16...17...18...19...20...21...22...23...24...25
.n
.2....4....4....6...10...14...20...30...44...60...60...28....2
.3....4....8...16...22...54...70..152..238..416..574..810..642..760..456..320..136...72....8
.4....4....8...20...40...84..126..268..418.1014.1450.2890.3510.5474.5286.7238.6926.8218.5636.6754.2956.4220..778...48
where k is the path length in nodes. There is insufficient evidence to attempt to define the irregularity of the array. However, the maximum values of k for 2 <= n <= 10 are 14, 20, 25, 30, 36, 42, 48, 53. Reading this array by rows gives the sequence. One half of the numbers of paths constitute the sequence to remove the effect of the bilateral symmetry of the rectangle.

			T(2,3) = One half of the number of complete non-self-adjacent simple paths of length 3 nodes within a square lattice bounded by a 2 X 9 node rectangle.

Cf. A213106, A213249, A213274, A213089, A213342, A213375, A213379, A213383, A213425.

cp's OEIS Frontend

A213342 Irregular array T(n,k) of numbers/2 of non-extendable non-self-adjacent simple paths of each length within a square lattice bounded by rectangles with nodal dimensions n and 4, n >= 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A213375 Irregular array T(n,k) of numbers/2 of non-extendable (complete) non-self-adjacent simple paths of each length within a square lattice bounded by rectangles with nodal dimensions n and 5, n >= 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A213954 Irregular array T(n,k) of the numbers of non-extendable (complete) non-self-adjacent simple paths starting at each of a minimal subset of nodes within a square lattice bounded by rectangles with nodal dimensions n and 3, n >= 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A214121 Irregular array T(n,k) of the numbers of non-extendable (complete) non-self-adjacent simple paths ending at each of a minimal subset of nodes within a square lattice bounded by rectangles with nodal dimensions n and 3, n >= 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A213379 Irregular array T(n,k) of numbers/2 of non-extendable (complete) non-self-adjacent simple paths of each length within a square lattice bounded by rectangles with nodal dimensions n and 6, n >= 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A214504 Irregular array T(n,k) of the numbers of non-extendable (complete) non-self-adjacent simple paths incorporating each of a minimal subset of nodes within a square lattice bounded by rectangles with nodal dimensions n and 3, n >= 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A213383 Irregular array T(n,k) of numbers/2 of non-extendable (complete) non-self-adjacent simple paths of each length within a square lattice bounded by rectangles with nodal dimensions n and 7, n >= 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A213425 Irregular array T(n,k) of numbers/2 of non-extendable (complete) non-self-adjacent simple paths of each length within a square lattice bounded by rectangles with nodal dimensions n and 8, n >= 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A213426 Irregular array T(n,k) of numbers/2 of non-extendable (complete) non-self-adjacent simple paths of each length within a square lattice bounded by rectangles with nodal dimensions n and 9, n >= 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs