A209656 -id:A209656 - cp's OEIS frontend

A209657 Meandric numbers for a river crossing up to 13 parallel roads at n points.

1, 1, 2, 4, 10, 23, 63, 155, 448, 1152, 3452, 9158, 28178, 76539, 240369, 665127, 2123408, 5964627, 19301713, 54897139, 179687084, 516448412, 1707004865, 4950415081, 16501638058, 48228801029, 161963084065, 476588705579, 1611037922998, 4769064680579, 16215807145689

Offset: 0

Robert Price, May 07 2012

nonn

Number of ways that a river (or directed line) that starts in the South and flows East can cross up to 13 parallel East-West roads n times.

Sequence derived from list of solutions described in A206432.

Andrew Howroyd, Table of n, a(n) for n = 0..40

Column k=13 of A380367.
Cf. A005316 (sequence for one road; extensive references and links).
Cf. A076876 (sequence for two parallel roads).
Cf. A204352, A208062, A208126, A208452, A208453, A209383, A209621, A209622, A209626, A209656, A209657, A209660, A209707, A210344, A210478, A210567, A210592 (sequences for 3 to 19 parallel roads).
Cf. A206432 (sequence for unlimited number of parallel roads).
Cf. A076875, A076906, A076907.

a(21) onwards from Andrew Howroyd, Jan 31 2025

A209660 Meandric numbers for a river crossing up to 14 parallel roads at n points.

Original entry on oeis.org

1, 1, 2, 4, 10, 23, 63, 155, 448, 1152, 3452, 9158, 28178, 76539, 240370, 665128, 2123437, 5964658, 19302248, 54897742, 179695133, 516457890, 1707112980, 4950547189, 16502992754, 48230509790, 161979310981, 476609746441, 1611226513378, 4769315213007, 16217954185533

Offset: 0

Robert Price, May 07 2012

nonn

Number of ways that a river (or directed line) that starts in the South and flows East can cross up to 14 parallel East-West roads n times.

Sequence derived from list of solutions described in A206432.

Andrew Howroyd, Table of n, a(n) for n = 0..40

Column k=14 of A380367.
Cf. A005316 (sequence for one road; extensive references and links).
Cf. A076876 (sequence for two parallel roads).
Cf. A204352, A208062, A208126, A208452, A208453, A209383, A209621, A209622, A209626, A209656, A209657, A209660, A209707, A210344, A210478, A210567, A210592 (sequences for 3 to 19 parallel roads).
Cf. A206432 (sequence for unlimited number of parallel roads).
Cf. A076875, A076906, A076907.

a(21) onwards from Andrew Howroyd, Jan 31 2025

A209707 Meandric numbers for a river crossing up to 15 parallel roads at n points.

Original entry on oeis.org

1, 1, 2, 4, 10, 23, 63, 155, 448, 1152, 3452, 9158, 28178, 76539, 240370, 665129, 2123438, 5964689, 19302281, 54898345, 179695808, 516467363, 1707124038, 4950679082, 16503152306, 48232213024, 161981435856, 476630676669, 1611253336964, 4769563838975, 16218280461398

Offset: 0

Robert Price, May 07 2012

nonn

Number of ways that a river (or directed line) that starts in the South and flows East can cross up to 15 parallel East-West roads n times.

Sequence derived from list of solutions described in A206432.

Andrew Howroyd, Table of n, a(n) for n = 0..40

Column k=15 of A380367.
Cf. A005316 (sequence for one road; extensive references and links).
Cf. A076876 (sequence for two parallel roads).
Cf. A204352, A208062, A208126, A208452, A208453, A209383, A209621, A209622, A209626, A209656, A209657, A209660, A209707, A210344, A210478, A210567, A210592 (sequences for 3 to 19 parallel roads).
Cf. A206432 (sequence for unlimited number of parallel roads).
Cf. A076875, A076906, A076907.

a(21) onwards from Andrew Howroyd, Jan 31 2025

A210344 Meandric numbers for a river crossing up to 16 parallel roads at n points.

Original entry on oeis.org

1, 1, 2, 4, 10, 23, 63, 155, 448, 1152, 3452, 9158, 28178, 76539, 240370, 665129, 2123439, 5964690, 19302314, 54898380, 179696483, 516468114, 1707135091, 4950691884, 16503311633, 48232404028, 161983554724, 476633293902, 1611280036392, 4769597694865, 16218604538134

Offset: 0

Robert Price, May 07 2012

nonn

Number of ways that a river (or directed line) that starts in the South and flows East can cross up to 16 parallel East-West roads n times.

Sequence derived from list of solutions described in A206432.

Andrew Howroyd, Table of n, a(n) for n = 0..40

Column k=16 of A380367.
Cf. A005316 (sequence for one road; extensive references and links).
Cf. A076876 (sequence for two parallel roads).
Cf. A204352, A208062, A208126, A208452, A208453, A209383, A209621, A209622, A209626, A209656, A209657, A209660, A209707, A210344, A210478, A210567, A210592 (sequences for 3 to 19 parallel roads).
Cf. A206432 (sequence for unlimited number of parallel roads).
Cf. A076875, A076906, A076907.

a(21) onwards from Andrew Howroyd, Jan 31 2025

A210478 Meandric numbers for a river crossing up to 17 parallel roads at n points.

Original entry on oeis.org

1, 1, 2, 4, 10, 23, 63, 155, 448, 1152, 3452, 9158, 28178, 76539, 240370, 665129, 2123439, 5964691, 19302315, 54898415, 179696520, 516468865, 1707135922, 4950704681, 16503326351, 48232594797, 161983781556, 476635904628, 1611283232139, 4769631412013, 16218646880868

Offset: 0

Robert Price, May 07 2012

nonn

Number of ways that a river (or directed line) that starts in the South and flows East can cross up to 17 parallel East-West roads n times.

Sequence derived from list of solutions described in A206432.

Andrew Howroyd, Table of n, a(n) for n = 0..40

Column k=17 of A380367.
Cf. A005316 (sequence for one road; extensive references and links).
Cf. A076876 (sequence for two parallel roads).
Cf. A204352, A208062, A208126, A208452, A208453, A209383, A209621, A209622, A209626, A209656, A209657, A209660, A209707, A210344, A210478, A210567, A210592 (sequences for 3 to 19 parallel roads).
Cf. A206432 (sequence for unlimited number of parallel roads).
Cf. A076875, A076906, A076907.

a(21) onwards from Andrew Howroyd, Jan 31 2025

A210567 Meandric numbers for a river crossing up to 18 parallel roads at n points.

Original entry on oeis.org

1, 1, 2, 4, 10, 23, 63, 155, 448, 1152, 3452, 9158, 28178, 76539, 240370, 665129, 2123439, 5964691, 19302316, 54898416, 179696557, 516468904, 1707136753, 4950705596, 16503341064, 48232611611, 161984008143, 476636172048, 1611286420859, 4769635283158, 16218689069176

Offset: 0

Robert Price, May 07 2012

nonn

Number of ways that a river (or directed line) that starts in the South and flows East can cross up to 18 parallel East-West roads n times.

Sequence derived from list of solutions described in A206432.

Andrew Howroyd, Table of n, a(n) for n = 0..40

Column k=18 of A380367.
Cf. A005316 (sequence for one road; extensive references and links).
Cf. A076876 (sequence for two parallel roads).
Cf. A204352, A208062, A208126, A208452, A208453, A209383, A209621, A209622, A209626, A209656, A209657, A209660, A209707, A210344, A210478, A210567, A210592 (sequences for 3 to 19 parallel roads).
Cf. A206432 (sequence for unlimited number of parallel roads).
Cf. A076875, A076906, A076907.

a(21) onwards from Andrew Howroyd, Jan 31 2025

A210592 Meandric numbers for a river crossing up to 19 parallel roads at n points.

Original entry on oeis.org

1, 1, 2, 4, 10, 23, 63, 155, 448, 1152, 3452, 9158, 28178, 76539, 240370, 665129, 2123439, 5964691, 19302316, 54898417, 179696558, 516468943, 1707136794, 4950706511, 16503342067, 48232628420, 161984027241, 476636439213, 1611286734027, 4769639146736, 16218693724179

Offset: 0

Robert Price, May 07 2012

nonn

Number of ways that a river (or directed line) that starts in the South and flows East can cross up to 19 parallel East-West roads n times.

Sequence derived from list of solutions described in A206432.

Andrew Howroyd, Table of n, a(n) for n = 0..40

Column k=19 of A380367.
Cf. A005316 (sequence for one road; extensive references and links).
Cf. A076876 (sequence for two parallel roads).
Cf. A204352, A208062, A208126, A208452, A208453, A209383, A209621, A209622, A209626, A209656, A209657, A209660, A209707, A210344, A210478, A210567, A210592 (sequences for 3 to 19 parallel roads).
Cf. A206432 (sequence for unlimited number of parallel roads).
Cf. A076875, A076906, A076907.

a(21) onwards from Andrew Howroyd, Jan 31 2025

A380367 Array read by antidiagonals: meandric numbers for a river crossing up to k parallel roads at n points, n >= 0, k >= 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 3, 3, 1, 1, 2, 4, 8, 8, 1, 1, 2, 4, 9, 14, 14, 1, 1, 2, 4, 10, 21, 43, 42, 1, 1, 2, 4, 10, 22, 52, 81, 81, 1, 1, 2, 4, 10, 23, 61, 131, 272, 262, 1, 1, 2, 4, 10, 23, 62, 142, 345, 538, 538, 1, 1, 2, 4, 10, 23, 63, 153, 420, 915, 1920, 1828

Offset: 0

Andrew Howroyd, Jan 31 2025

Illustrations of the initial terms for the case of two parallel roads can be found in A076876.

			Array begins:
===================================================
n\k |   1   2   3    4    5    6    7    8    9 ...
----+----------------------------------------------
  0 |   1   1   1    1    1    1    1    1    1 ...
  1 |   1   1   1    1    1    1    1    1    1 ...
  2 |   1   2   2    2    2    2    2    2    2 ...
  3 |   2   3   4    4    4    4    4    4    4 ...
  4 |   3   8   9   10   10   10   10   10   10 ...
  5 |   8  14  21   22   23   23   23   23   23 ...
  6 |  14  43  52   61   62   63   63   63   63 ...
  7 |  42  81 131  142  153  154  155  155  155 ...
  8 |  81 272 345  420  433  446  447  448  448 ...
  9 | 262 538 915 1017 1120 1135 1150 1151 1152 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..945 (first 43 antidiagonals)

Columns 1..19 are A005316, A076876, A204352, A208062, A208126, A208452, A208453, A209383, A209621, A209622, A209626, A209656, A209657, A209660, A209707, A210344, A210478, A210567, A210592.
Main diagonal is A206432.
Cf. A076875 (perpendicular roads).

T(n,k) = T(n,n) for k > n.

A209918 Tetrahedron in which the n-th slice is also one of the three views of the shell model of partitions of A207380 with n shells.

Original entry on oeis.org

1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 5, 4, 2, 1, 1, 2, 2, 1, 1, 1, 7, 6, 4, 2, 1, 1, 2, 3, 2, 1, 1, 2, 1, 1, 1

Offset: 1

Omar E. Pol, Mar 26 2012

Each slice of the tetrahedron is a triangle, thus the number of elements in the n-th slice is A000217(n). The slices are perpendicular to the slices of A026792. Each element of the n-th slice equals the volume of a column of the shell model of partitions with n shells. The sum of each column of the n-th slice is A000041(n). The sum of all elements of the n-th slice is A066186(n).
It appears that the triangle formed by the first row of each slice gives A058399.
It appears that the triangle formed by the last column of each slice gives A008284 and A058398.
Also consider a vertical rectangle on the infinite square grid with shorter side = n and longer side = p(n) = A000041(n). Each row of rectangle represents a partition of n. Each part of each partition of n is a horizontal rectangle with shorter side = 1 and longer side = k, where k is the size of the part. It appears that T(n,k,j) is also the number of k-th parts of all partitions of n in the j-th column of rectangle.

			---------------------------------------------------------
Illustration of first five                       A181187
slices of the tetrahedron                        Row sum
---------------------------------------------------------
. 1,                                                1
.    2, 1,                                          3
.       1,                                          1
.          3, 2, 1                                  6
.             1, 1,                                 2
.                1,                                 1
.                   5, 4, 2, 1,                    12
.                      1, 2, 2,                     5
.                         1, 1                      2
.                            1,                     1
.                               7, 6, 4, 2, 1,     20
.                                  1, 2, 3, 2,      8
.                                     1, 1, 2,      4
.                                        1, 1,      2
.                                           1,      1
--------------------------------------------------------
. 1, 2, 2, 3, 3, 3, 5, 5, 5, 5, 7, 7, 7, 7, 7,
.
Note that the 5th slice appears as one of three views of the model in the example section of A207380.

Row sums give A181187. Column sums give A209656. Main diagonal gives A210765. Another version is A209655.

Cf. A000041, A000217, A002260, A004736, A008284, A026792, A058398, A058399, A066186, A135010, A182703, A182715, A207380.

A210765 Triangle read by rows in which row n lists the number of partitions of n together with n-1 ones.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 5, 1, 1, 1, 7, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 15, 1, 1, 1, 1, 1, 1, 22, 1, 1, 1, 1, 1, 1, 1, 30, 1, 1, 1, 1, 1, 1, 1, 1, 42, 1, 1, 1, 1, 1, 1, 1, 1, 1, 56, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 77, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 101, 1, 1, 1

Offset: 1

Omar E. Pol, Mar 26 2012

The sum of row n is S_n = n - 1 + A000041(n) = A133041(n) - 1.

Also consider a vertical rectangle on the infinite square grid with shorter side = n and longer side = p(n) = A000041(n). Each row of rectangle represents a partition of n. Each part of each partition of n is a horizontal rectangle with shorter side = 1 and longer side = k, where k is the size of the part. It appears that T(n,k) is also the number of k-th parts of all partitions of n in the k-th column of rectangle.

			Triangle begins:
1;
2,  1;
3,  1, 1;
5,  1, 1, 1;
7,  1, 1, 1, 1;
11, 1, 1, 1, 1, 1;
15, 1, 1, 1, 1, 1, 1;
22, 1, 1, 1, 1, 1, 1, 1;
30, 1, 1, 1, 1, 1, 1, 1, 1;
42, 1, 1, 1, 1, 1, 1, 1, 1, 1;

Main diagonal of A209655 and of A209918.

Cf. A000041, A008284, A058399, A133041, A135010, A141285, A209656, A207380.

cp's OEIS Frontend

A209657 Meandric numbers for a river crossing up to 13 parallel roads at n points.

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A209660 Meandric numbers for a river crossing up to 14 parallel roads at n points.

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A209707 Meandric numbers for a river crossing up to 15 parallel roads at n points.

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A210344 Meandric numbers for a river crossing up to 16 parallel roads at n points.

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A210478 Meandric numbers for a river crossing up to 17 parallel roads at n points.

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A210567 Meandric numbers for a river crossing up to 18 parallel roads at n points.

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A210592 Meandric numbers for a river crossing up to 19 parallel roads at n points.

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A380367 Array read by antidiagonals: meandric numbers for a river crossing up to k parallel roads at n points, n >= 0, k >= 1.

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Formula

A209918 Tetrahedron in which the n-th slice is also one of the three views of the shell model of partitions of A207380 with n shells.

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A210765 Triangle read by rows in which row n lists the number of partitions of n together with n-1 ones.

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