A213478 -id:A213478 - cp's OEIS frontend

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.

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.

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

A214399 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.

Original entry on oeis.org

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

Offset: 2

Christopher Hunt Gribble, Jul 15 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......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.

			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.

Cf. A213106, A213249, A213274, A213478, A214119, A214397.

Corrected by Christopher Hunt Gribble, Jul 19 2012

A214023 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 5, n >= 2.

Original entry on oeis.org

8, 7, 6, 36, 25, 20, 32, 18, 12, 122, 102, 94, 110, 52, 32, 436, 395, 394, 395, 220, 154, 394, 154, 80, 1580, 1414, 1402, 1381, 813, 596, 1365, 652, 432, 5600, 4829, 4650, 4795, 2792, 2036, 4453, 2285, 1712, 4412, 2556, 2248, 19287, 16131, 15246, 16735, 9444, 6758, 15113, 7697, 5858, 13878, 8612, 8496

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 3 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........8.....7.....6
..3.......36....25....20....32....18....12
..4......122...102....94...110....52....32
..5......436...395...394...395...220...154...394...154....80
..6.....1580..1414..1402..1381...813...596..1365...652...432
..7.....5600..4829..4650..4795..2792..2036..4453..2285..1712..4412..2556..2248
..8....19287.16131.15246.16735..9444..6758.15113..7697..5858.13878..8612..8496
where k indicates the position of the start node in the quarter-rectangle.
For each n, the maximum value of k is 3*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 8 9
NT 8 7 6 7 8
   8 7 6 7 8
To limit duplication, only the top left-hand corner 8 and the 7 and 6 to its right are stored in the sequence, i.e. T(2,1) = 8, T(2,2) = 7 and T(2,3) = 6.

Cf. A213106, A213249, A213375, A213478, A213954, A214022

A214025 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 6, n >= 2.

Original entry on oeis.org

13, 10, 8, 77, 51, 38, 68, 36, 20, 330, 266, 248, 300, 145, 96, 1580, 1381, 1365, 1414, 813, 652, 1402, 596, 432, 7678, 6630, 6357, 6630, 3968, 3192, 6357, 3192, 2828, 35971, 30070, 27638, 30709, 18037, 13744, 27591, 14507, 13851, 26574, 15318, 17846

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 3 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.......13....10.....8
..3.......77....51....38....68....36....20
..4......330...266...248...300...145....96
..5.....1580..1381..1365..1414...813...652..1402...596...432
..6.....7678..6630..6357..6630..3968..3192..6357..3192..2828
..7....35971.30070.27638.30709.18037.13744.27591.14507.13851.26574.15318.17846
where k indicates the position of the start node in the quarter-rectangle.
For each n, the maximum value of k is 3*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  8  9 10 11
NT 13 10  8  8 10 13
   13 10  8  8 10 13
To limit duplication, only the top left-hand corner 13 and the 10 and 8 to its right are stored in the sequence, i.e. T(2,1) = 13, T(2,2) = 10 and T(2,3) = 8.

Cf. A213106, A213249, A213375, A213478, A213954, A214022, A214023

A214037 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 7, n >= 2.

Original entry on oeis.org

21, 15, 11, 10, 164, 106, 72, 64, 142, 72, 38, 28, 888, 695, 607, 602, 780, 385, 258, 270, 5600, 4795, 4453, 4412, 4829, 2792, 2285, 2556, 4650, 2036, 1712, 2248, 35971, 30709, 27591, 26574, 30070, 18037, 14507, 15318, 27638, 13744, 13851, 17846

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 4 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.......21....15....11....10
.3......164...106....72....64....142...72....38....28
.4......888...695...607...602...780...385...258...270
.5.....5600..4795..4453..4412..4829..2792..2285..2556..4650..2036..1712..2248
.6....35971.30709.27591.26574.30070.18037.14507.15318.27638.13744.13851.17846
where k indicates the position of the start node in the quarter-rectangle.
For each n, the maximum value of k is 4*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  8  9 10 11 12 13
NT 21 15 11 10 11 15 21
   21 15 11 10 11 15 21
To limit duplication, only the top left-hand corner 21 and the 15, 11 and 10 to its right are stored in the sequence, i.e. T(2,1) = 21, T(2,2) = 15, T(2,3) = 11 and T(2,4) = 10.

Cf. A213106, A213249, A213379, A213478, A213954, A214022, A214023, A214025

A214038 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 8, n >= 2.

Original entry on oeis.org

34, 23, 16, 13, 347, 225, 142, 109, 298, 146, 74, 46, 2347, 1842, 1526, 1387, 2008, 1001, 663, 669, 19287, 16735, 15113, 13878, 6131, 9444, 7697, 8612, 15246, 6758, 5858, 8496, 163666, 141849, 126129, 112049, 132636, 81112, 65551, 67006, 118724, 58677, 60918, 87046

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 4 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.......34.....23.....16.....13
.3......347....225....142....109....298....146.....74.....46
.4.....2347...1842...1526...1387...2008...1001....663....669
.5....19287..16735..15113..13878...6131...9444...7697...8612..15246...6758...5858...8496
.6...163666.141849.126129.112049.132636..81112..65551..67006.118724..58677..60918..87046
where k indicates the position of the start node in the quarter-rectangle.
For each n, the maximum value of k is 4*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
    8  9 10 11 12 13 14 15
NT 34 23 16 13 13 16 23 34
   34 23 16 13 13 16 23 34
To limit duplication, only the top left-hand corner 34 and the 23, 16 and 13 to its right are stored in the sequence, i.e. T(2,1) = 34, T(2,2) = 23, T(2,3) = 16 and T(2,4) = 13.

Cf. A213106, A213249, A213425, A213478, A213954, A214022, A214023, A214025, A214037

A214042 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 9, n >= 2.

Original entry on oeis.org

55, 36, 24, 18, 16, 732, 476, 294, 197, 168, 628, 302, 148, 82, 64, 6115, 4840, 3979, 3349, 3076, 5170, 2597, 1718, 1595, 1564, 64904, 57210, 52820, 46787, 43294, 53478, 31544, 26459, 28472, 28700, 50228, 22432, 19802, 27924, 30696

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 5 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....13....14....15
.n
.2......55....36....24....18....16
.3.....732...476...294...197...168...628...302...148....82....64
.4....6115..4840..3979..3349..3076..5170..2597..1718..1595..1564
.5...64904.57210.52820.46787.43294.53478.31544.26459.28472.28700.50228.22432.19802.27924.30696
where k indicates the position of the start node in the quarter-rectangle.
For each n, the maximum value of k is 5*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  8
    9 10 11 12 13 14 15 16 17
NT 55 36 24 18 16 18 24 36 55
   55 36 24 18 16 18 24 36 55
To limit duplication, only the top left-hand corner 34 and the 23, 16 and 13 to its right are stored in the sequence, i.e. T(2,1) = 34, T(2,2) = 23, T(2,3) = 16 and T(2,4) = 13.

Cf. A213106, A213249, A213426, A213478, A213954, A214022, A214023, A214025, A214037, A214038

Comment corrected by Christopher Hunt Gribble, Jul 22 2012

cp's OEIS Frontend

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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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.

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A214399 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.

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A214023 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 5, n >= 2.

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A214025 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 6, n >= 2.

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A214037 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 7, n >= 2.

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A214038 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 8, n >= 2.

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A214042 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 9, n >= 2.

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