A000136 -id:A000136 - cp's OEIS frontend

A223094 Number of foldings of n labeled stamps in which leaf n is inwards.

0, 0, 2, 6, 26, 78, 288, 888, 3130, 9850, 34112, 108998, 374636, 1211046, 4148816, 13533796, 46304730, 152153758, 520434552, 1720325302, 5885686496, 19552190624, 66927118548, 223264746520, 764725528072, 2560239468774, 8775478294368, 29470844083770

Offset: 1

N. J. A. Sloane, Mar 29 2013

nonn

Subset of foldings of n labeled stamps (A000136). [Stéphane Legendre, Apr 09 2013]
From Roger Ford, Aug 23 2024: (Start)
a(n) represents the number of impossible stamp foldings with stamp 1 on top and n+1 stamps that are correctly folded for the first n stamps. From stamp n to stamp n+1, the stamp connection crosses a folding so the folding is impossible.
Example a(3) = 2.  Impossible foldings = 1,3,2,4 and 1,4,2,3.
               1 __                      1 __
Stamp numbers  3 __|__   Vertical Lines  4 __|__
               2 |_|  |  lines are folds 2 __| |
               4 _____|                  3 |___|
a(4) = 6, and that means for 5 stamps there are 6 impossible foldings with the first impossible folding occurring from stamp 4 to stamp 5.  Impossible foldings = 1,2,4,3,5; 1,2,5,3,4; 1,3,4,2,5; 1,4,3,5,2; 1,5,2,4,3; 1,5,3,4,2. (End)

Stéphane Legendre, Table of n, a(n) for n = 1..43
Stéphane Legendre, Foldings and Meanders, arXiv preprint arXiv:1302.2025 [math.CO], 2013.
Index entries for sequences obtained by enumerating foldings

Cf. A000136, A000682.

A000682 = Import["https://oeis.org/A000682/b000682.txt", "Table"][[All, 2]];
a[n_] := n A000682[[n]] - A000682[[n + 1]];
Array[a, 43] (* Jean-François Alcover, Sep 02 2019 *)

a(n) = A000136(n) - A000682(n+1). - Andrew Howroyd, Dec 05 2015

For n >= 3: a(n) = n! - Sum_{k=3..n-1} (a(k)*n!/k!) - A000682(n+1). - Roger Ford, Aug 24 2024

More terms from Stéphane Legendre, Apr 09 2013

A223095 Number of foldings of n labeled stamps in which both end leaves are inwards.

Original entry on oeis.org

0, 0, 0, 2, 10, 40, 156, 546, 1986, 6716, 23742, 79472, 277178, 925588, 3205896, 10711486, 36963722, 123712788, 426075994, 1429030624, 4916833424, 16526958144, 56840484232, 191466923584, 658460090994, 2222507917328, 7644360501390, 25850724646008, 88938175307354

Offset: 1

N. J. A. Sloane, Mar 29 2013

nonn

Subset of foldings of n labeled stamps (A000136). - Stéphane Legendre, Apr 09 2013

Stéphane Legendre, Table of n, a(n) for n = 1..42
S. Legendre, Foldings and Meanders, arXiv preprint arXiv:1302.2025 [math.CO], 2013.
S. Legendre, Foldings and Meanders, Aust. J. Comb. 58(2), 275-291, 2014.
Index entries for sequences obtained by enumerating foldings

Cf. A000136, A000682, A077014.

Cf. A223093, A223094.

A217318 = Import["https://oeis.org/A217318/b217318.txt", "Table"][[All, 2]];
a[n_] := (2 - Mod[n, 2]) A217318[[n]];
Array[a, 42] (* Jean-François Alcover, Sep 02 2019 *)

a(n) = A223094(n) - A223093(n). - Andrew Howroyd, Dec 06 2015
a(n) = A000136(n) + A077014(n) - 2 * A000682(n). - Andrew Howroyd, Dec 06 2015
A217318(n) = a(n) if n is odd and A217318(n) = (1/2)*a(n) if n is even. - Stéphane Legendre, Jan 13 2014

Name clarified by Stéphane Legendre, Apr 09 2013

More terms from Stéphane Legendre, Apr 09 2013

A227167 The number of meandering curves of order n.

Original entry on oeis.org

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

A330269 The number of semi-meanders with n top arches and concentric arches within the starting arch or a starting arch with length one.

Original entry on oeis.org

1, 1, 2, 4, 8, 18, 42, 108, 282, 786, 2192, 6402, 18600, 55978, 167256, 514102, 1567976, 4896164, 15170630, 47957260, 150468678, 480371736, 1522649458, 4900568718, 15665593150, 50761432998, 163431901126, 532624603680, 1725349278270, 5650796083020, 18401781369182

Offset: 1

Roger Ford, Dec 07 2019

nonn

			For n = 5, a(5) = 8:
        /\                                  /\
       //\\                      /\        /  \          /\
      ///\\\      /\  /\        /  \      /  /\\    /\  //\\
   /\////\\\\, /\//\\//\\, /\/\//\/\\, /\//\//\\\, //\\///\\\,
                              /\
                 /\          //\\          starting arch
    /\  /\      //\\  /\    ///\\\        (1) (2) (3) (4)
   //\\//\\/\, ///\\\//\\, ////\\\\/\,     4 + 2 + 1 + 1 = 8.

Jean-François Alcover, Table of n, a(n) for n = 1..46

Cf. A000682.

A000136 = Import["https://oeis.org/A000136/b000136.txt", "Table"][[All, 2]];
a[n_] := If[n == 1, 1, Sum[A000136[[k]]/k, {k, 1, n - 1}]];
a /@ Range[46] (* Jean-François Alcover, Feb 13 2020 *)

a(1) = 1, for n >= 2, a(n) = Sum_{k=1..n-1} A000682(k).

More terms from Jinyuan Wang, Dec 08 2019

A001416 Number of ways of folding a 3 X n strip of stamps.

Original entry on oeis.org

1, 6, 60, 1368, 15552, 201240, 2016432, 21582624, 201060768, 1944012744, 17257455960, 156760071600, 1346073913440, 11734199738820, 98420246759688

Offset: 0

N. J. A. Sloane

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robert Dickau, Stamp Folding, 2010.
Hunter Hogan, mapFolding: A computational framework for enumerating distinct folding patterns of rectangular maps, GitHub repository, (2025).
W. F. Lunnon, Multi-dimensional map-folding, The Computer Journal, Volume 14, Issue 1, 1971, Pages 75-80.
Eric Weisstein's World of Mathematics, Map Folding.
Index entries for sequences obtained by enumerating foldings

Cf. A000136, A000560, A001415, A001417, A001418, A195646.

a(8)-a(14) from Fred Lunnon and Sean A. Irvine, Dec 26 2017

cp's OEIS Frontend

A223094 Number of foldings of n labeled stamps in which leaf n is inwards.

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A223095 Number of foldings of n labeled stamps in which both end leaves are inwards.

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A227167 The number of meandering curves of order n.

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A330269 The number of semi-meanders with n top arches and concentric arches within the starting arch or a starting arch with length one.

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A001416 Number of ways of folding a 3 X n strip of stamps.

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