A046726 -id:A046726 - cp's OEIS frontend

A046721 Number of semi-meanders of order n with 2 components.

1, 2, 6, 16, 48, 140, 428, 1308, 4072, 12796, 40432, 129432, 413900, 1342580, 4335288, 14201804, 46226896, 152594276, 500016036, 1660630740, 5472190206, 18264517264, 60475691308, 202684618564, 673892675030, 2266436498400, 7562707682032, 25510762766704, 85394319699916

Offset: 2

N. J. A. Sloane

nonn

Andrew Howroyd, Table of n, a(n) for n = 2..40
P. Di Francesco, O. Golinelli and E. Guitter, Meander, folding and arch statistics, arXiv:hep-th/9506030, 1995.

Cf. A000682, A046726.

a(15)-a(30) from Andrew Howroyd, Nov 27 2015

A046722 Number of semi-meanders of order n with 3 components.

Original entry on oeis.org

1, 3, 11, 37, 126, 430, 1454, 4976, 16880, 57824, 197010, 675428, 2310268, 7927778, 27205180, 93448486, 321537086, 1105589516, 3812424912, 13121988240, 45330375774, 156172996170, 540314673678, 1863197292582, 6454265995454, 22275589419432, 77246945788890, 266813803179348

Offset: 3

N. J. A. Sloane

nonn

Andrew Howroyd, Table of n, a(n) for n = 3..40
P. Di Francesco, O. Golinelli, and E. Guitter, Meander, folding and arch statistics, arXiv:hep-th/9506030, 1995.
P. Di Francesco, O. Golinelli, and E. Guitter, Meander, folding and arch statistics, Mathematical and Computer Modelling 26 (1997), 97-147.

Cf. A000682, A046726.

a(15)-a(30) from Andrew Howroyd, Nov 27 2015

A046723 Number of semi-meanders of order n with 4 components.

Original entry on oeis.org

1, 4, 17, 66, 254, 956, 3584, 13256, 49052, 179552, 658560, 2394504, 8724464, 31575096, 114451388, 412811544, 1490190544, 5360943684, 19288139802, 69245171564, 248463024330, 890477645192, 3188033497580, 11409453277272, 40771092374710, 145735210316376, 519955750491512

Offset: 4

N. J. A. Sloane

nonn

Andrew Howroyd, Table of n, a(n) for n = 4..40
P. Di Francesco, O. Golinelli and E. Guitter, Meander, folding and arch statistics, arXiv:hep-th/9506030, 1995.

Cf. A000682, A046726.

a(15)-a(40) from Andrew Howroyd, Dec 07 2015

A046724 Number of semi-meanders of order n with 5 components.

Original entry on oeis.org

1, 5, 24, 104, 438, 1796, 7238, 28848, 113518, 444278, 1720384, 6643492, 25421620, 97136712, 368280210, 1395104236, 5250325378, 19746342212, 73863421894, 276113486146, 1027609657470, 3821478801772, 14161346139866, 52428406903688, 193568833452364, 713860635606784

Offset: 5

N. J. A. Sloane

nonn

Andrew Howroyd, Table of n, a(n) for n = 5..40
P. Di Francesco, O. Golinelli and E. Guitter, Meander, folding and arch statistics, arXiv:hep-th/9506030, 1995.

Cf. A000682, A046726.

a(15)-a(40) from Andrew Howroyd, Dec 07 2015

A046725 Number of semi-meanders of order n with 6 components.

Original entry on oeis.org

1, 6, 32, 152, 690, 3028, 12996, 54812, 228284, 939148, 3833076, 15487428, 62244564, 247973928, 984221764, 3876113404, 15223550024, 59379645924, 231124139318, 894157177372, 3453279084296, 13266154255196, 50886266714598, 194294744477756, 740816697816046

Offset: 6

N. J. A. Sloane

nonn

Andrew Howroyd, Table of n, a(n) for n = 6..40
P. Di Francesco, O. Golinelli and E. Guitter, Meander, folding and arch statistics, arXiv:hep-th/9506030, 1995.

Cf. A000682, A046726.

a(15)-a(40) from Andrew Howroyd, Dec 07 2015

A265507 A pyramid T(n,p,k) of square arrays read by rows relating semimeanders(n), positive arches(p) and components(k).

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 0, 1, 0, 5, 0, 0, 2, 0, 4, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 8, 0, 1, 0, 9, 0, 10, 0, 2, 0, 8, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 13, 0, 0, 1, 0, 13, 0, 36, 0, 0, 3, 0, 23, 0, 24, 0, 0, 3, 0, 12, 0

Offset: 1

Roger Ford, Dec 09 2015

A positive arch is defined as a top arch that starts at an odd-numbered vertex and ends at a higher even-numbered vertex.
For each value of n there is a square array with n^2 elements.
Rows are in order of decreasing number of components.
The sum of all the elements in each square array(n) = Catalan numbers C(n) A000108.
The sum of columns for array(n) = Semimeander components row(n) A046726.
The sum of the rows for array(n) = Narayana numbers T(n,k) A001263.
All semimeander solutions (k=1) for array n have positive arches = floor((n+2)/2).

			For n=3:                                   /\          /\
               /\               /\        /  \        //\\
              /  \             /  \      /    \      //  \\
  /\ /\ /\   / /\ \  /\   /\  / /\ \    //\  /\\    // /\ \\
  \ \\// /   \ \ \/ / /   \ \ \/ / /    \\ \/ //    \\ \/ //
   \ \/ /     \ \  / /     \ \  / /      \\  //      \\  //
    \  /       \ \/ /       \ \/ /        \\//        \\//
     \/         \  /         \  /          \/          \/
                 \/           \/
  p=3,k=2     p=2,k=1      p=2,k=1      p=1,k=2     p=2,k=3.
.
n=3  p\k 3  2  1   n=9  p\k 9  8  7  6  5  4  3  2  1
      1: 0  1  0         1: 0  0  0  0  1  0  0  0  0
      2: 1  0  2         2: 0  0  0  4  0 32  0  0  0
      3: 0  1  0         3: 0  0  6  0 78  0 252 0  0
                         4: 0  4  0 72  0 446 0 654 0
                         5: 1  0 29  0 280 0 950 0 504
                         6: 0  4  0 72  0 446 0 654 0
                         7: 0  0  6  0 78  0 252 0  0
                         8: 0  0  0  4  0 32  0  0  0
                         9: 0  0  0  0  1  0  0  0  0

Cf. A000108, A001263, A046726.

cp's OEIS Frontend

A046721 Number of semi-meanders of order n with 2 components.

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A046722 Number of semi-meanders of order n with 3 components.

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A046723 Number of semi-meanders of order n with 4 components.

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A046724 Number of semi-meanders of order n with 5 components.

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A046725 Number of semi-meanders of order n with 6 components.

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A265507 A pyramid T(n,p,k) of square arrays read by rows relating semimeanders(n), positive arches(p) and components(k).

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