A213106 -id:A213106 - cp's OEIS frontend

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.

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

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

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.

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

A214122 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 4, n >= 2.

Original entry on oeis.org

10, 0, 33, 6, 4, 0, 90, 22, 22, 4, 256, 52, 67, 14, 88, 32, 720, 104, 187, 30, 236, 108, 1931, 200, 495, 56, 622, 262, 602, 364, 5029, 386, 1245, 106, 1624, 618, 1537, 898, 12996, 744, 3061, 206, 4080, 1502, 3938, 2186, 3744, 2196, 33512, 1422, 7615, 398

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
..n
..2......10.....0
..3......33.....6.....4.....0
..4......90....22....22.....4
..5.....256....52....67....14....88....32
..6.....720...104...187....30...236...108
..7....1931...200...495....56...622...262...602...364
..8....5029...386..1245...106..1624...618..1537...898
..9...12996...744..3061...206..4080..1502..3938..2186..3744..2196
.10...33512..1422..7615...398.10014..3676..9775..5466..9177..5246
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  6  7
NT 10  0  0 10
   10  0  0 10
To limit duplication, only the top left-hand corner 10 and the 0 to its right are stored in the sequence, i.e. T(2,1) = 10 and T(2,2) = 0.

Cf. A213106, A213249, A213342, A214022, A214119, A214121.

Comment corrected by Christopher Hunt Gribble, Jul 22 2012

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

A213431 Irregular array T(n,k) of the numbers of distinct shapes under rotation of the 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

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

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
..n
..2....2
..3....2...4...2
..4....2...4...6...6
..5....2...4...6..10..10...2
..6....2...4...6..10..14..16...8
..7....2...4...6..10..14..20..26..18...2
..8....2...4...6..10..14..20..30..40..34..10
..9....2...4...6..10..14..20..30..44..60..60..28...2
.10....2...4...6..10..14..20..30..44..64..90.100..62..12
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 n + floor((n+1)/2) for n >= 2. Reading this array by rows gives the sequence.

			T(2,3) = The number of distinct shapes under rotation of the 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 distinct shapes under rotation of the complete non-self-adjacent simple paths of each nodal length k for n >> 0 appears to be 2*A097333(2:), that is, 2*(Sum(j=0..k-2, C(k-2-j, floor(j/2)))), for k >= 4.

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

A214359 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 5, n >= 2.

Original entry on oeis.org

18, 0, 0, 75, 13, 16, 6, 0, 0, 256, 67, 88, 52, 14, 32, 932, 246, 308, 246, 80, 130, 308, 130, 288, 3431, 746, 920, 992, 251, 352, 1179, 580, 1210, 12027, 2143, 2612, 3522, 640, 954, 4399, 1941, 3956, 4170, 2394, 5136, 40489, 6345, 7544, 11359, 1689, 2772, 15642, 6165, 12824, 15239, 8214, 16728

Offset: 2

Christopher Hunt Gribble, Jul 13 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......18.....0.....0
.3......75....13....16.....6.....0.....0
.4.....256....67....88....52....14....32
.5.....932...246...308...246....80...130...308...130...288
.6....3431...746...920...992...251...352..1179...580..1210
.7...12027..2143..2612..3522...640...954..4399..1941..3956..4170..2394..5136
.8...40489..6345..7544.11359..1689..2772.15642..6165.12824.15239..8214.16728
where k indicates the position of the end 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 end node (EN) of a complete non-self-adjacent simple path is
EN  0  1  2  3  4
    5  6  7  8  9
NT 18  0  0  0 18
   18  0  0  0 18
To limit duplication, only the top left-hand corner 18 and the two zeros to its right are stored in the sequence, i.e. T(2,1) = 10, T(2,2) = 0 and T(2,3) = 0.

Cf. A213106, A213249, A213375, A214023, A214119, A214121, A214122.

Comment corrected by Christopher Hunt Gribble, Jul 22 2012

A214510 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 4, n >= 2.

Original entry on oeis.org

23, 24, 80, 86, 88, 100, 264, 303, 303, 282, 820, 1008, 1007, 907, 1058, 776, 2401, 3043, 3013, 2844, 3312, 2375, 6751, 8651, 8562, 8317, 9411, 7116, 9718, 6882, 18630, 24035, 23979, 23261, 26077, 20216, 26479, 20016, 50775, 65977, 66474, 63790, 72137, 55400, 71469, 55907, 69764, 57274

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......23....24
.3......80....86....88...100
.4.....264...303...303...282
.5.....820..1008..1007...907..1058...776
.6....2401..3043..3013..2844..3312..2375
.7....6751..8651..8562..8317..9411..7116..9718..6882
.8...18630.24035.23979.23261.26077.20216.26479.20016
.9...50775.65977.66474.63790.72137.55400.71469.55907.69764.57274
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  6  7
NT 23 24 24 23
   23 24 24 23
To limit duplication, only the top left-hand corner 23 and the 24 to its right are stored in the sequence,
i.e. T(2,1) = 23 and T(2,2) = 24.

Cf. A213106, A213249, A213342, A214022, A214122, A214397, A214399, A214504

Comment corrected by Christopher Hunt Gribble, Jul 22 2012

cp's OEIS Frontend

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.

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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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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.

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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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A214122 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 4, n >= 2.

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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.

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A213431 Irregular array T(n,k) of the numbers of distinct shapes under rotation of the 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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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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A214359 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 5, n >= 2.

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