Antonios Panayotopoulos - cp's OEIS frontend

A227167 The number of meandering curves of order n.

1, 1, 6, 8, 50, 72, 462, 696, 4536, 7030, 46310, 73188, 485914, 778946, 5202690, 8430992, 56579196, 92470194, 622945970, 1025114180, 6927964218, 11465054942, 77692142980, 129180293184, 877395996200, 1464716085664, 9968202968958, 16698145444260, 113837957337750, 191264779292430

Offset: 1

Antonios Panayotopoulos, Panayotis Vlamos, Jul 03 2013

nonn

A meandering curve of order n is a continuous curve which does not intersect itself yet intersects a horizontal line n times.

The set of meandering curves of order n is partitioned into the following three classes: curves with no extremities (A005316), curves with only one extremity (A217310), and curves with both extremities covered by their arcs (A217318).

A. Panayotopoulos and P. Tsikouras, Properties of meanders, JCMCC 46 (2003), 181-190.

Jean-François Alcover, Table of n, a(n) for n = 1..45
J. E. Koehler, Folding a strip of stamps, J. Combin. Theory, 5 (1968), 135-152.
W. F. Lunnon, A map-folding problem, Math. Comp. 22 (1968), 193-199.
A. Panayotopoulos, P. Vlamos, Partitioning the Meandering Curves, Mathematics in Computer Science (2015) p 1-10.

A000136 = Cases[Import["https://oeis.org/A000136/b000136.txt", "Table"], {, }][[All, 2]];
a[n_] := If[OddQ[n], A000136[[n]], A000136[[n]]/2];
a /@ Range[1, 45] (* Jean-François Alcover, Sep 20 2019 *)

a(n) = A000136(n) if n is odd and a(n) = (1/2)*A000136(n) if n is even.

a(n) = A217310(n) + A217318(n) + A005316(n). - Andrew Howroyd, Dec 07 2015

A217310 The number of meandering curves of order n, with only one extremity covered by its arcs.

Original entry on oeis.org

0, 0, 4, 4, 32, 38, 264, 342, 2288, 3134, 20740, 29526, 194916, 285458, 1885840, 2822310, 18682016, 28440970, 188717116, 291294678, 1937706144, 3025232480, 20173268632, 31797822936, 212530874156, 337731551446, 2262235585956, 3620119437762, 24297593488468

Offset: 1

Panayotis Vlamos, Antonios Panayotopoulos, Georgia Theocharopoulou, Mar 17 2013

nonn

A meandering curve of order n is a continuous curve which does not intersect itself yet intersects a horizontal line n times.

A. Panayotopoulos and P. Tsikouras, Properties of meanders, JCMCC 46 (2003), 181-190.

Panayotis Vlamos, Table of n, a(n) for n = 1..42
A. Panayotopoulos, P. Vlamos, Partitioning the Meandering Curves, Mathematics in Computer Science (2015) p 1-10.

Cf. A005315.

a(n) = A223093(n) * A000034(n). - Andrew Howroyd, Dec 06 2015

A217318 The number of meandering curves of order n, with both extremity covered by its arcs.

Original entry on oeis.org

0, 0, 0, 1, 10, 20, 156, 273, 1986, 3358, 23742, 39736, 277178, 462794, 3205896, 5355743, 36963722, 61856394, 426075994, 714515312, 4916833424, 8263479072, 56840484232, 95733461792, 658460090994, 1111253958664, 7644360501390, 12925362323004, 88938175307354

Offset: 1

Panayotis Vlamos, Antonios Panayotopoulos, and Georgia Theocharopoulou, Mar 18 2013

nonn

A meandering curve of order n is a continuous curve which does not intersect itself yet intersects a horizontal line n times.

A. Panayotopoulos and P. Tsikouras, Properties of meanders, JCMCC 46 (2003), 181-190.

Panayotis Vlamos, Table of n, a(n) for n = 1..42
A. Panayotopoulos, P. Vlamos, Partitioning the Meandering Curves, Mathematics in Computer Science (2015) p 1-10.

Cf. A005315.

a(n) = A223095(n) * A000034(n) / 2. - Andrew Howroyd, Dec 06 2015

A208357 Number of meanders of order 2n+1 (4*n+2 crossings of the infinite line) with central 1-1 cut.

Original entry on oeis.org

4, 64, 1764, 68644, 3341584, 190992400, 12310790116, 871343837764, 66469126179600, 5391179227622500, 460213149486493456, 41024422751464102500, 3795407861954983718544, 362631040029370613957184, 35638591665642822414493156, 3590789985613539065908070116, 369893506453438150061450367376

Offset: 1

Panayotis Vlamos and Antonios Panayotopoulos, Feb 25 2012

nonn

Antonios Panayotopoulos and Panos Tsikouras, Properties of meanders, JCMCC 46 (2003), 181-190.
Antonios Panayotopoulos and Panayiotis Vlamos, Meandric Polygons, Ars Combinatoria 87 (2008), 147-159.

S. K. Lando and A. K. Zvonkin, Plane and projective meanders, Theoretical Computer Science Vol. 117, pp. 227-241, 1993.
Antonios Panayotopoulos and Panos Tsikouras, The multimatching property of nested sets, Math. & Sci. Hum. 149 (2000), 23-30.
Antonios Panayotopoulos and Panos Tsikouras, Meanders and Motzkin Words, J. Integer Seq., Vol. 7 (2004), Article 04.1.2.
Antonios Panayotopoulos and Panayiotis Vlamos, Cutting Degree of Meanders, Artificial Intelligence Applications and Innovations, IFIP Advances in Information and Communication Technology, Volume 382, 2012, pp 480-489; DOI 10.1007/978-3-642-33412-2_49. - From _N. J. A. Sloane_, Dec 29 2012

Cf. A005315, A192927, A207851.

a(n) = A005315(n+1)^2.

More terms using the data at A005315 added by Amiram Eldar, Jun 09 2024

A207851 Number of meanders of order 2n+1 (4*n+2 crossings of the infinite line) with only central 1-1 cut (no other 1-1 cuts).

Original entry on oeis.org

4, 16, 324, 12100, 595984, 35236096, 2363709924, 174221090404, 13815880848784, 1161868621405636, 102544273501721104, 9424551852935116804, 896612457556434503824, 87881363502264179831824, 8840846163309028336017124

Offset: 1

Panayotis Vlamos and Antonios Panayotopoulos, Feb 21 2012

nonn

Central cut is a 1-1 cut at the center of the meander (the i-line is for i=n).

A. Panayotopoulos and P. Tsikouras, Properties of meanders, JCMCC 46 (2003), 181-190.
A. Panayotopoulos and P. Vlamos, Meandric Polygons, Ars Combinatoria 87 (2008), 147-159.

Panayotis Vlamos, Table of n, a(n) for n = 1..22
Iwan Jensen, Enumeration of plane meanders, arXiv:cond-mat/9910313 [cond-mat.stat-mech], 1999.
S. K. Lando and A. K. Zvonkin, Plane and projective meanders, Theoretical Computer Science Vol. 117, pp. 227-241, 1993.
A. Panayotopoulos and P. Tsikouras, The multimatching property of nested sets, Math. & Sci. Hum. 149 (2000), 23-30.
A. Panayotopoulos and P. Tsikouras, Meanders and Motzkin Words, J. Integer Seqs., Vol. 7, 2004.
A. Panayotopoulos and P. Vlamos, Cutting Degree of Meanders, Artificial Intelligence Applications and Innovations, IFIP Advances in Information and Communication Technology, Volume 382, 2012, pp 480-489; DOI 10.1007/978-3-642-33412-2_49. - From _N. J. A. Sloane_, Dec 29 2012

Cf. A005315, A192927.

A203019 Number of elevated peakless Motzkin paths.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 4, 8, 17, 37, 82, 185, 423, 978, 2283, 5373, 12735, 30372, 72832, 175502, 424748, 1032004, 2516347, 6155441, 15101701, 37150472, 91618049, 226460893, 560954047, 1392251012, 3461824644, 8622571758, 21511212261, 53745962199

Offset: 0

Panayotis Vlamos and Antonios Panayotopoulos, Dec 27 2011

nonn

Essentially the same as A004148: a(0)=a(1)=0 and a(n) = A004148(n-2) for n>=2.

			G.f. = x^2 + x^3 + x^4 + 2*x^5 + 4*x^6 + 8*x^7 + 17*x^8 + 37*x^9 + ...

A. Panayotopoulos and P. Tsikouras, Properties of meanders, JCMCC 46 (2003), 181-190.
A. Panayotopoulos and P. Vlamos, Meandric Polygons, Ars Combinatoria 87 (2008), 147-159.

Muniru A Asiru, Table of n, a(n) for n = 0..300
Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
I. Jensen, Enumeration of plane meanders, arXiv:cond-mat/9910313 [cond-mat.stat-mech], 1999.
S. K. Lando and A. K. Zvonkin, Plane and projective meanders, Theoretical Computer Science Vol. 117 (1993) p. 232.
A. Panayotopoulos and P. Tsikouras, The multimatching property of nested sets, Math. & Sci. Hum. 149 (2000), 23-30.
A. Panayotopoulos and P. Tsikouras, Meanders and Motzkin Words, J. Integer Seqs., Vol. 7, 2004.
A. Panayotopoulos and P. Vlamos, Cutting Degree of Meanders, Artificial Intelligence Applications and Innovations, IFIP Advances in Information and Communication Technology, Volume 382, 2012, pp 480-489; DOI 10.1007/978-3-642-33412-2_49. - From _N. J. A. Sloane_, Dec 29 2012

List([0..40],n->Sum([0..Int((n-1)/2)],m->Binomial(2*m+1,m)*Sum([0..n-2*m-2],k->(Binomial(k,n-2*m-k-2)*Binomial(2*m+k,k)*(-1)^(n-k))/(2*m+1)))); # Muniru A Asiru, Aug 13 2018

terms = 34;
A[] = 0; Do[A[x] = x (x - A[x] / (A[x] - 1)) + O[x]^terms, {terms}];
CoefficientList[A[x], x] (* Jean-François Alcover, Jul 27 2018, after Michael Somos *)
Table[Sum[Binomial[2*m + 1, m]*Sum[(Binomial[k, n - 2*m - k - 2]* Binomial[2*m + k, k]*(-1)^(n - k))/(2*m + 1), {k, 0, n - 2*m - 2}], {m, 0, Floor[(n - 1)/2]}], {n, 0, 50}] (* G. C. Greubel, Aug 12 2018 *)

a(n):=sum((binomial(2*m+1,m)*sum(binomial(k,n-2*m-k-2)*binomial(2*m+k,k)*(-1)^(n-k),k,0,n-2*m-2))/(2*m+1),m,0,(n-1)/2); /* Vladimir Kruchinin, Mar 12 2016 */

{a(n) = local(A); A = O(x); for( k=1, ceil(n / 3), A = x^2 / (1 - x / (1 - A))); polcoeff( A, n)} /* Michael Somos, May 12 2012 */

G.f.: x^2 / (1 - x / (1 - x^2 / (1 - x / (1 - x^2 / (1 - x / (1 - x^2 / ...)))))). - Michael Somos, May 12 2012
G.f. A(x) =: y satisfies y / x = x + y / (1 - y). - Michael Somos, Jan 31 2014
G.f. A(x) =: y satisfies y = x^2 + (x - x^2)*y + y*y. - Michael Somos, Jan 31 2014
Given g.f. A(x), then B(x) = A(x)/x satisfies B(-B(-x)) = x. - Michael Somos, Jan 31 2014
a(n) = Sum_{m=0..(n-1)/2}((binomial(2*m+1,m)*Sum_{k=0..n-2*m-2}(binomial(k,n-2*m-k-2)*binomial(2*m+k,k)*(-1)^(n-k)))/(2*m+1)). - Vladimir Kruchinin, Mar 12 2016
a(n) ~ 5^(1/4) * phi^(2*n - 2) / (2*sqrt(Pi)*n^(3/2)), where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Aug 14 2018
D-finite with recurrence n*a(n) +(-2*n+3)*a(n-1) +(-n+3)*a(n-2) +(-2*n+9)*a(n-3) +(n-6)*a(n-4)=0. - R. J. Mathar, Jan 25 2023

A192927 Number of meanders of order n, with 1-1 cut.

Original entry on oeis.org

0, 0, 4, 24, 152, 1056, 7884, 62336, 516060, 4435888, 39338456, 358164768, 3335087752, 31663393880, 305740631660, 2996450625216, 29756512979124, 298991371001192, 3036071594865808, 31123847225593944, 321822051245586716, 3353854189297573504, 35203483037473883368, 371948773494408980616, 3953755503217954761364

Offset: 1

Panayotis Vlamos and Antonios Panayotopoulos, Aug 03 2011

nonn

S. K. Lando and A. K. Zvonkin, Plane and projective meanders, Séries Formelles et Combinatoire Algebrique. Laboratoire Bordelais de Recherche Informatique, Universite Bordeaux I, 1991, pp. 287-303.

Iwan Jensen, Enumeration of plane meander, arXiv:cond-mat/9910313v1 [cond-mat.stat-mech], 1999.
I. Jensen, A transfer matrix approach to the enumeration of plane meanders, J. Phys. A 33, 5953-5963 (2000).
A. Panayotopoulos and P. Tsikouras, Meanders and Motzkin Words, J. Integer Seqs., Vol. 7, 2004.
A. Panayotopoulos and P. Vlamos, Cutting Degree of Meanders, Artificial Intelligence Applications and Innovations, IFIP Advances in Information and Communication Technology, Volume 382, 2012, pp 480-489; DOI 10.1007/978-3-642-33412-2_49. - From _N. J. A. Sloane_, Dec 29 2012

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User: Antonios Panayotopoulos

A227167 The number of meandering curves of order n.

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A217310 The number of meandering curves of order n, with only one extremity covered by its arcs.

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A217318 The number of meandering curves of order n, with both extremity covered by its arcs.

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A208357 Number of meanders of order 2n+1 (4*n+2 crossings of the infinite line) with central 1-1 cut.

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A207851 Number of meanders of order 2n+1 (4*n+2 crossings of the infinite line) with only central 1-1 cut (no other 1-1 cuts).

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A203019 Number of elevated peakless Motzkin paths.

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A192927 Number of meanders of order n, with 1-1 cut.

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