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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 3, n >= 2.
+10
9
4, 4, 2, 4, 8, 12, 0, 8, 4, 8, 16, 18, 14, 8, 14, 4, 8, 16, 22, 42, 24, 42, 22, 18, 4, 8, 16, 22, 48, 60, 82, 90, 66, 34, 24, 2, 4, 8, 16, 22, 50, 66, 132, 160, 218, 120, 122, 56, 36, 4, 4, 8, 16, 22, 52, 68, 144, 222, 334, 406, 302, 288, 198, 88, 52, 6, 4, 8, 16, 22, 54, 70, 152, 238, 416, 574, 810, 642, 760, 456, 320, 136, 72, 8
COMMENTS
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...2
.3....4...8..12...0...8
.4....4...8..16..18..14...8..14
.5....4...8..16..22..42..24..42..22..18
.6....4...8..16..22..48..60..82..90..66..34..24...2
.7....4...8..16..22..50..66.132.160.218.120.122..56..36...4
.8....4...8..16..22..52..68.144.222.334.406.302.288.198..88..52...6
.9....4...8..16..22..54..70.152.238.416..74.810.642.760.456.320.136..72...8
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 2n+1 for 2 <= n <= 6 and 2n+2 for n >= 7. 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.
EXAMPLE
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 3 node rectangle.
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 2, n >= 2.
+10
9
2, 3, 4, 5, 5, 8, 7, 6, 13, 10, 8, 21, 15, 11, 10, 34, 23, 16, 13, 55, 36, 24, 18, 16, 89, 57, 37, 26, 21, 144, 91, 58, 39, 29, 26, 233, 146, 92, 60, 42, 34, 377, 235, 147, 94, 63, 47, 42, 610, 379, 236, 149, 97, 68, 55, 987, 612, 380, 238, 152, 102, 76, 68, 1597, 989, 613, 382, 241, 157, 110, 89
COMMENTS
The subset of nodes approximately defines the top left-hand quarter of the rectangle and has nodal dimensions floor((n+1)/2) and 1 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
..n
..2.......2
..3.......3...4
..4.......5...5
..5.......8...7...6
..6......13..10...8
..7......21..15..11..10
..8......34..23..16..13
..9......55..36..24..18..16
.10......89..57..37..26..21
.11.....144..91..58..39..29..26
.12.....233.146..92..60..42..34
.13.....377.235.147..94..63..47..42
.14.....610.379.236.149..97..68..55
.15.....987.612.380.238.152.102..76..68
.16....1597.989.613.382.241.157.110..89
where k indicates the position of the start node in the quarter-rectangle. For each n, the maximum value of k is floor((n+1)/2). Reading this array by rows gives the sequence.
FORMULA
Let T(n,k) denote an element of the irregular array then it appears that
EXAMPLE
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
NT 2 2
2 2
To limit duplication, only the top left-hand corner 2 is stored in the sequence, i.e. T(2,1) = 2.
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.
+10
8
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
COMMENTS
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.
EXAMPLE
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.
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.
+10
8
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
COMMENTS
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.
EXAMPLE
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.
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 2, n >= 2.
+10
8
2, 5, 0, 10, 0, 18, 0, 0, 31, 0, 0, 52, 0, 0, 0, 86, 0, 0, 0, 141, 0, 0, 0, 0, 230, 0, 0, 0, 0, 374, 0, 0, 0, 0, 0, 607, 0, 0, 0, 0, 0, 984, 0, 0, 0, 0, 0, 0, 1594, 0, 0, 0, 0, 0, 0, 2581, 0, 0, 0, 0, 0, 0, 0, 4178, 0, 0, 0, 0, 0, 0, 0, 6762, 0, 0, 0, 0, 0, 0
COMMENTS
The subset of nodes is contained in the top left-hand quarter of the rectangle and has nodal dimensions floor((n+1)/2) and 1 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.......2
..3.......5..0
..4......10..0
..5......18..0..0
..6......31..0..0
..7......52..0..0..0
..8......86..0..0..0
..9.....141..0..0..0..0
.10.....230..0..0..0..0
.11.....374..0..0..0..0..0
.12.....607..0..0..0..0..0
.13.....984..0..0..0..0..0..0
.14....1594..0..0..0..0..0..0
.15....2581..0..0..0..0..0..0..0
.16....4178..0..0..0..0..0..0..0
.17....6762..0..0..0..0..0..0..0..0
.18...10943..0..0..0..0..0..0..0..0
.19...17708..0..0..0..0..0..0..0..0..0
.20...28654..0..0..0..0..0..0..0..0..0
where k indicates the position of the end node in the quarter-rectangle. For each n, the maximum value of k is floor((n+1)/2). Reading this array by rows gives the sequence.
FORMULA
Let T(n,k) denote an element of the irregular array then it appears that T(n,k) = A000045(n+3) - 3, n >= 2, k = 1 and T(n,k) = 0, n >= 2, k >= 2.
EXAMPLE
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
NT 2 2
2 2
To limit duplication, only the top left-hand corner 2 is stored in the sequence, i.e. T(2,1) = 2.
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 2, n >= 2.
+10
7
6, 12, 14, 23, 24, 40, 42, 40, 68, 70, 70, 113, 116, 116, 122, 186, 190, 192, 202, 304, 310, 314, 334, 334, 495, 504, 512, 546, 552, 804, 818, 832, 890, 902, 912, 1304, 1326, 1350, 1446, 1470, 1490, 2113, 2148, 2188, 2346, 2388, 2428, 2434, 3422, 3478, 3544, 3802, 3874, 3944, 3966
COMMENTS
The subset of nodes is contained in the top left-hand quarter of the rectangle and has nodal dimensions floor((n+1)/2) and 1 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......6
..3.....12...14
..4.....23...24
..5.....40...42...40
..6.....68...70...70
..7....113..116..116..122
..8....186..190..192..202
..9....304..310..314..334..334
.10....495..504..512..546..552
.11....804..818..832..890..902..912
.12...1304.1326.1350.1446.1470.1490
.13...2113.2148.2188.2346.2388.2428.2434
.14...3422.3478.3544.3802.3874.3944.3966
.15...5540.5630.5738.6158.6278.6398.6442.6462
where k indicates the position of a node in the quarter-rectangle.
For each n, the maximum value of k is floor((n+1)/2).
Reading this array by rows gives the sequence.
EXAMPLE
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
NT 6 6
6 6
To limit duplication, only the top left-hand corner 6 is stored in the sequence, i.e. T(2,1) = 6.
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.
+10
6
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
COMMENTS
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.
EXAMPLE
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.
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.
+10
4
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
COMMENTS
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.
EXAMPLE
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.
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.
+10
4
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
COMMENTS
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.
EXAMPLE
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.
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.
+10
3
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
COMMENTS
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.
EXAMPLE
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.
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