A213106 -id:A213106 - cp's OEIS frontend

A213249 Triangle T(n,k) of numbers of distinct shapes under rotation of non-extendable (complete) non-self-adjacent simple paths within a square lattice bounded by rectangles with nodal dimensions n and k, n >= k >= 2.

2, 8, 16, 18, 64, 134, 34, 170, 706, 1854, 60, 398, 2346, 13198, 41478, 102, 880, 6832, 55454, 382116, 1424988

Offset: 2

Christopher Hunt Gribble, Jun 07 2012

The triangle of numbers is:
....k....2....3.....4......5.......6........7
.n
.2.......2
.3.......8...16
.4......18...64...134
.5......34..170...706...1854
.6......60..398..2346..13198...41478
.7.....102..880..6832..55454..382116..1424988
The sequence is formed by reading the triangle by rows.

			T(2,2) = The number of rotationally distinct complete non-self-adjacent simple path shapes within a 2 X 2 node rectangle.

Cf. A213106.

Let T(n,k) denote an element of the triangle then the following recurrence relations appear to hold:
T(n, 2) - T(n-1, 2) - 2*A000045(n+1) = 0, n >= 3,
T(n, 3) - 2*T(n-1, 3) - T(n-4, 3) - 4*(n+11) = 0, n >= 7.

A213274 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 2, n >= 2.

Original entry on oeis.org

4, 4, 4, 2, 4, 4, 6, 6, 4, 4, 6, 10, 10, 2, 4, 4, 6, 10, 14, 16, 8, 4, 4, 6, 10, 14, 20, 26, 18, 2, 4, 4, 6, 10, 14, 20, 30, 40, 34, 10, 4, 4, 6, 10, 14, 20, 30, 44, 60, 60, 28, 2, 4, 4, 6, 10, 14, 20, 30, 44, 64, 90, 100, 62, 12, 4, 4, 6, 10, 14, 20, 30, 44, 64, 94, 134, 160, 122, 40, 2

Offset: 2

Christopher Hunt Gribble, Jun 08 2012

The irregular array of numbers is:
....k..3...4...5...6...7...8...9..10..11..12..13..14..15..16..17
..n
..2....4
..3....4...4...2
..4....4...4...6...6
..5....4...4...6..10..10...2
..6....4...4...6..10..14..16...8
..7....4...4...6..10..14..20..26..18...2
..8....4...4...6..10..14..20..30..40..34..10
..9....4...4...6..10..14..20..30..44..60..60..28...2
.10....4...4...6..10..14..20..30..44..64..90.100..62..12
.11....4...4...6..10..14..20..30..44..64..94.134.160.122..40...2
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 + n mod 2)/2 for n >= 2.
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 a 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 2 node rectangle.

Cf. A213106, A213249.

The asymptotic sequence for the number of paths of each nodal length k for n >> 0 appears to be 2*A097333(1:), that is, 2*(Sum(j=0..k-2, C(k-2-j, floor(j/2)))), for k >= 3.

A213089 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.

Original entry on oeis.org

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

Offset: 2

Christopher Hunt Gribble, Jun 08 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...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.

			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.

Cf. A213106, A213249, A213274.

A213478 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.

Original entry on oeis.org

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

Offset: 2

Christopher Hunt Gribble, Jun 12 2012

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.

			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.

Cf. A213106, A213249, A213274.

Let T(n,k) denote an element of the irregular array then it appears that
T(n,k) = A000045(n-k+2), k = 0
T(n,k) = A000045(n-k+2) + A000045(k+1), k > 0.

Improved Comments

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.

Original entry on oeis.org

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

A214119 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.

Original entry on oeis.org

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

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 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.

			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.

Cf. A213106, A213249, A213274, A213478.

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.

A214397 Triangle T(n,k) of the numbers of nodes in all non-extendable (complete) non-self-adjacent simple paths within a square lattice bounded by rectangles with nodal dimensions n and k, n >= k >= 2.

Original entry on oeis.org

24, 76, 320, 188, 1040, 4608, 408, 2756, 18636, 104272, 832, 8368, 67952, 513460, 3349208, 1624, 21468, 228432, 2312112, 19845964, 152434216, 3080, 53108, 730772, 9943160, 113061272, 1125079096, 10676325280, 5716, 128072, 2261792, 41508164, 629214072, 8150708696, 99701732480, 1200653865056

Offset: 2

Christopher Hunt Gribble, Jul 15 2012

The triangle of numbers is:
....k.....2......3.......4........5.........6..........7...........8.............9
.n
.2.......24
.3.......76....320
.4......188...1040....4608
.5......408...2756...18636...104272
.6......832...8368...67952...513460...3349208
.7.....1624..21468..228432..2312112..19845964..152434216
.8.....3080..53108..730772..9943160.113061272.1125079096.10676325280
.9.....5716.128072.2261792.41508164.629214072.8150708696.99701732480.1200653865056
Reading this triangle by rows gives the sequence.

			T(2,2) = The number of nodes in all complete non-self-adjacent simple paths within a 2 X 2 node rectangle.

Cf. A213106, A213249.

A214022 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 4, n >= 2.

Original entry on oeis.org

5, 5, 17, 12, 14, 10, 46, 37, 37, 18, 122, 110, 102, 52, 94, 32, 330, 300, 266, 145, 248, 96, 888, 780, 695, 385, 607, 258, 602, 270, 2347, 2008, 1842, 1001, 1526, 663, 1387, 669, 6115, 5170, 4840, 2597, 3979, 1718, 3349, 1595, 3076, 1564, 15811, 13288, 12545, 6722, 10331, 4481, 8461, 3925, 7181, 3556

Offset: 2

Christopher Hunt Gribble, Jul 01 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........5.....5
..3.......17....12....14....10
..4.......46....37....37....18
..5......122...110...102....52....94....32
..6......330...300...266...145...248....96
..7......888...780...695...385...607...258...602...270
..8.....2347..2008..1842..1001..1526...663..1387...669
..9.....6115..5170..4840..2597..3979..1718..3349..1595..3076..1564
.10....15811.13288.12545..6722.10331..4481..8461..3925..7181..3556
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 6 7
NT 5 5 5 5
   5 5 5 5
To limit duplication, only the top left-hand corner 5 and the 5 to its right are stored in the sequence, i.e. T(2,1) = 5 and T(2,2) = 5.

Cf. A213106, A213249, A213342, A213478, A213954

cp's OEIS Frontend

A213249 Triangle T(n,k) of numbers of distinct shapes under rotation of non-extendable (complete) non-self-adjacent simple paths within a square lattice bounded by rectangles with nodal dimensions n and k, n >= k >= 2.

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A213274 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 2, n >= 2.

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A213089 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.

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A213478 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.

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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.

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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.

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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.

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A214119 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.

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A214397 Triangle T(n,k) of the numbers of nodes in all non-extendable (complete) non-self-adjacent simple paths within a square lattice bounded by rectangles with nodal dimensions n and k, n >= k >= 2.

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A214022 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 4, n >= 2.