A001997 -id:A001997 - cp's OEIS frontend

A000682 Semi-meanders: number of ways a semi-infinite directed curve can cross a straight line n times.

1, 1, 2, 4, 10, 24, 66, 174, 504, 1406, 4210, 12198, 37378, 111278, 346846, 1053874, 3328188, 10274466, 32786630, 102511418, 329903058, 1042277722, 3377919260, 10765024432, 35095839848, 112670468128, 369192702554, 1192724674590, 3925446804750

Offset: 1

N. J. A. Sloane

For n > 1, the number of permutations of n letters without overlaps [Sade, 1949]. - N. J. A. Sloane, Jul 05 2015
Number of ways to fold a strip of n labeled stamps with leaf 1 on top. [Clarified by Stéphane Legendre, Apr 09 2013]
From Roger Ford, Jul 04 2014: (Start)
The number of semi-meander solutions for n (a(n)) is equal to the number of n top arch solutions in the intersection of A001263 (with no intersecting top arches) and A244312 (arches forming a complete loop).
The top and bottom arches for semi-meanders pass through vertices 1-2n on a straight line with the arches below the line forming a rainbow pattern.
The number of total arches going from an odd vertex to a higher even vertex must be exactly 2 greater than the number of arches going from an even vertex to a higher odd vertex to form a single complete loop with no intersections.
The arch solutions in the intersection of A001263 (T(n,k)) and A244312 (F(n,k)) occur when the number of top arches going from an odd vertex to a higher even vertex (k) meets the condition that k = ceiling((n+1)/2).
Example: semi-meanders a(5)=10.
(A244312) F(5,3)=16 { 10 common solutions: [12,34,5 10,67,89] [16,23,45,78,9 10] [12,36,45,7 10,89] [14,23,58,67,9 10] [12,3 10,49,58,67] [18,27,36,45,9 10] [12,3 10,45,69,78] [18,25,34,67,9 10] [14,23,5 10,69,78] [16,25,34,7 10,89] } + [18,27,34,5 10,69] [16,25,3 10,49,78] [18,25,36,49,7 10] [14,27,3 10,58,69] [14,27,36,5 10,89] [16,23,49,58,7 10]
(A001263) T(5,3)=20  { 10 common solutions } + [12,38,45,67,9 10] [1 10,29,38,47,56] [1 10,25,34,69,78] [14,23,56,7 10,89] [12,3 10,47,56,89] [18,23,47,56,9 10] [1 10,29,36,45,78] [1 10,29,34,58,67] [1 10,27,34,56,89] [1 10,23,49,56,78].
(End)
From Roger Ford, Feb 23 2018: (Start)
For n>1, the number of semi-meanders with n top arches and k concentric starting arcs is a(n,k)= A000682(n-k).
                             /\          /\
Examples:  a(5,1)=4         //\\        /  \          /\
     A000682(5-1)=4        ///\\\      /  /\\        /  \       /\  /\
                        /\////\\\\, /\//\//\\\, /\/\//\/\\, /\ //\\//\\
           a(5,2)=2        /\                      a(5,3)=1   /\
     A000682(5-2)=2   /\  //\\    /\  /\     A000682(5-3)=1  //\\  /\
                     //\\///\\\, //\\//\\/\                 ///\\\//\\
           a(5,4)=1     /\
     A000682(5-4)=1    //\\
                      ///\\\
                     ////\\\\/\.   (End)
For n >= 4, 4*a(n-2) is the number of stamp foldings with leaf 1 on top, with leaf 2 in the second or n-th position, and with leaf n and leaf n-1 adjacent. Example for n = 5, 4*a(5-2) = 8: 12345, 12354, 12453, 12543, 13452, 13542, 14532, 15432. - Roger Ford, Aug 05 2019
From Martin Philp, Mar 25 2021: (Start)
The condition of having leaf n and leaf n-1 adjacent is the same as having one fewer leaf, and then counting each element twice. So the above comment is equivalent to saying:
For n >= 3, 2*a(n-1) is the number of stamp foldings with leaf 1 on top and leaf 2 in the second or n-th position. Example for n = 4, 2*a(4-1) = 4: 1234, 1243, 1342, 1432. Furthermore the number of stamp foldings with leaf 1 on top and leaf 2 in the n-th position is the same as the number of stamp foldings with leaf 1 on top and leaf 2 in the second position, as a cyclic rotation of 1 and mirroring the sequence maps one to the other. 1234, 1243 <-rot-> 2341, 2431 <-mirror-> 1432, 1342.
Hence, for n >= 2, a(n-1) is the number of stamp foldings having 1 and 2 (in this order) on top.
Not only is a(n) the number of stamp foldings with 1 on top, it is the number of stamp foldings with any particular leaf on top. This explains why A000136(n)= n*a(n).
(End)
The number of semi-meanders that in the first exterior top arch has exactly one arch of length one = Sum_{k=1..n-1} a(k).  Example: for n = 5, Sum_{k=1..4} A000682(k) = 8, 10 = arch of length one, *start and end of first exterior top arch*; *10*11001100, *10*11110000, *10*11011000, *10*10110100, *1100*111000, *1100*110010, *111000*1100, *11110000*10. - Roger Ford, Jul 12 2020

			a(4) = 4: the four solutions with three crossings are the two solutions shown in A086441(3) together with their reflections about a North-South axis.

A. Sade, Sur les Chevauchements des Permutations, published by the author, Marseille, 1949.
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).

I. Jensen, Table of n, a(n) for n = 1..45
CombOS - Combinatorial Object Server, Generate meanders and stamp foldings
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, Meanders: a direct enumeration approach, arXiv:hep-th/9607039, 1996; Nucl. Phys. B 482 [ FS ] (1996) 497-535.
P. Di Francesco, Matrix model combinatorics: applications to folding and coloring, arXiv:math-ph/9911002, 1999.
I. Jensen, Home page
I. Jensen, Terms a(1)..a(45)
I. Jensen, A transfer matrix approach to the enumeration of plane meanders, J. Phys. A 33, 5953-5963 (2000).
I. Jensen and A. J. Guttmann, Critical exponents of plane meanders J. Phys. A 33, L187-L192 (2000).
J. E. Koehler, Folding a strip of stamps, J. Combin. Theory, 5 (1968), 135-152.
J. E. Koehler, Folding a strip of stamps, J. Combin. Theory, 5 (1968), 135-152. [Annotated, corrected, scanned copy]
M. La Croix, Approaches to the Enumerative Theory of Meanders
Stéphane Legendre, Foldings and Meanders, arXiv preprint arXiv:1302.2025 [math.CO], 2013.
Stéphane Legendre, Illustration of initial terms
Bowie Liu, Dennis Wong, Chan-Tong Lam, and Marcus Im, Recursive and iterative approaches to generate rotation Gray codes for stamp foldings and semi-meanders, arXiv:2411.05458 [cs.DS], 2024. See p. 2.
W. F. Lunnon, A map-folding problem, Math. Comp. 22 (1968), 193-199.
A. Panayotopoulos and P. Vlamos, Partitioning the Meandering Curves, Mathematics in Computer Science (2015) p 1-10.
Albert Sade, Sur les Chevauchements des Permutations, published by the author, Marseille, 1949. [Annotated scanned copy]
J. Sawada and R. Li, Stamp foldings, semi-meanders, and open meanders: fast generation algorithms, Electronic Journal of Combinatorics, Volume 19 No. 2 (2012), P#43 (16 pages).
J. Touchard, Contributions à l'étude du problème des timbres poste, Canad. J. Math., 2 (1950), 385-398.
Index entries for sequences obtained by enumerating foldings

Cf. A000136, A001011, A001997, A000560 (nonisomorphic), A086441.

Row sums of A259689.

A000136 = Import["https://oeis.org/A000136/b000136.txt", "Table"][[All, 2]];
a[n_] := A000136[[n]]/n;
Array[a, 45] (* Jean-François Alcover, Sep 06 2019 *)

a(n) = 2*A000560(n-1) for n >= 3.
For n >= 2, a(n) = 2^(n-2) + Sum_{x=3..n-2} (2^(n-x-2)*A301620(x)). - Roger Ford, Apr 23 2018
a(n) = 2^(n-2) + Sum_{j=4..n-1} (Sum_{k=3..floor((j+2)/2)} (A259689(j,k)*(k-2)*2^(n-1-j))). - Roger Ford, Dec 12 2018
a(n) = A000136(n)/n. - Jean-François Alcover, Sep 06 2019, from formula in A000136.
a(n) = (n-1)! - Sum_{k=3..n-1} (A223094(k) * (n-1)! / k!). - Roger Ford, Aug 23 2024

Sade gives the first 11 terms. Computed to n = 45 by Iwan Jensen.

Offset changed by Roger Ford, Feb 09 2018

A001998 Bending a piece of wire of length n+1; walks of length n+1 on a tetrahedron; also non-branched catafusenes with n+2 condensed hexagons.

Original entry on oeis.org

1, 2, 4, 10, 25, 70, 196, 574, 1681, 5002, 14884, 44530, 133225, 399310, 1196836, 3589414, 10764961, 32291602, 96864964, 290585050, 871725625, 2615147350, 7845353476, 23535971854, 70607649841, 211822683802, 635467254244, 1906400965570, 5719200505225, 17157599124190

Offset: 0

N. J. A. Sloane

The wire stays in the plane, there are n bends, each is R,L or O; turning the wire over does not count as a new figure.
Equivalently, walks of n+1 steps on a tetrahedron, visiting n+2 vertices, with n "corners"; the symmetry group is S4, reversing a walk does not count as different. Simply interpret R,L,O as instructions to turn R, turn L, or retrace the last step. Walks are not self-avoiding.
Also, it appears that a(n) gives the number of equivalence classes of n-tuples of 0, 1 and 2, where two n-tuples are equivalent if one can be obtained from the other by a sequence of operations R and C, where R denotes reversal and C denotes taking the 2's complement (C(x)=2-x). This has been verified up to a(19)=290585050. Example: for n=3 there are ten equivalence classes {000, 222}, {001, 100, 122, 221}, {002, 022, 200, 220}, {010, 212}, {011, 110, 112, 211}, {012, 210}, {020, 202}, {021, 102, 120, 201}, {101, 121}, {111}, so a(3)=10. - John W. Layman, Oct 13 2009
There exists a bijection between chains of n+2 hexagons and the above described equivalence classes of n-tuples of 0,1, and 2. Namely, for a given chain of n+2 hexagons we take the sequence of the numbers of vertices of degree 2 (0, 1, or 2) between the consecutive contact vertices on one side of the chain; switching to the other side we obtain the 2's complement of this sequence; reversing the order of the hexagons, we obtain the reverse sequence. The inverse mapping is straightforward. For example, to a linear chain of 7 hexagons there corresponds the 5-tuple 11111. - Emeric Deutsch, Apr 22 2013
If we treat two wire bends (or walks, or tuples) related by turning over (or reversing) as different in any of the above-given interpretations of this sequence, we get A007051 (or A124302). Also, a(n-1) is the sum of first 3 terms in n-th row of A284949, see crossrefs therein. - Andrey Zabolotskiy, Sep 29 2017
a(n-1) is the number of color patterns (set partitions) in an unoriented row of length n using 3 or fewer colors (subsets). - Robert A. Russell, Oct 28 2018
From Allan Bickle, Jun 02 2022: (Start)
a(n) is the number of (unlabeled) 3-paths with n+6 vertices. (A 3-path with order n at least 5 can be constructed from a 4-clique by iteratively adding a new 3-leaf (vertex of degree 3) adjacent to an existing 3-clique containing an existing 3-leaf.)
Recurrences appear in the papers by Bickle, Eckhoff, and Markenzon et al. (End)
a(n) is also the number of distinct planar embeddings of the (n+1)-alkane graph (up to at least n=9, and likely for all n). - Eric W. Weisstein, May 21 2024

			There are 2 ways to bend a piece of wire of length 2 (bend it or not).
For n=4 and a(n-1)=10, the 6 achiral patterns are AAAA, AABB, ABAB, ABBA, ABCA, and ABBC.  The 4 chiral pairs are AAAB-ABBB, AABA-ABAA, AABC-ABCC, and ABAC-ABCB. - _Robert A. Russell_, Oct 28 2018

A. T. Balaban, Enumeration of Cyclic Graphs, pp. 63-105 of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976; see p. 75.
S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, Enumeration of tree-like octagonal systems: catapolyoctagons, ACH Models in Chem. 134 (1997), 55-70.
M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2.]
R. C. Read, The Enumeration of Acyclic Chemical Compounds, pp. 25-61 of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976. [I think this reference does not mention this sequence. - N. J. A. Sloane, Aug 10 2006]
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).

Indranil Ghosh, Table of n, a(n) for n = 0..2092 (terms 0..500 from T. D. Noe)
A. T. Balaban, J. Brunvoll, B. N. Cyvin, and S. J. Cyvin, Enumeration of branched catacondensed benzenoid hydrocarbons and their numbers of Kekulé structures, Tetrahedron 44(1) (1988), 221-228. See Eq. (6), p. 223.
A. T. Balaban and F. Harary, Chemical graphs V: enumeration and proposed nomenclature of benzenoid cata-condensed polycyclic aromatic hydrocarbons, Tetrahedron 24 (1968), 2505-2516.
Christian Barrientos and Sarah Minion, On the Graceful Cartesian Product of Alpha-Trees, Theory and Applications of Graphs, Vol. 4: Iss. 1, Article 3, 2017. (It mentions this sequence on p. 7.)
L. W. Beineke and R. E. Pippert, On the enumeration of planar trees of hexagons, Glasgow Math. J., 15 (1974), 131-147.
L. W. Beineke and R. E. Pippert, On the enumeration of planar trees of hexagons, Glasgow Math. J., 15 (1974), 131-147 [annotated scanned copy].
Allan Bickle, How to Count k-Paths, J. Integer Sequences, 25 (2022) Article 22.5.6.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, Isomer enumeration of some polygonal systems representing polycyclic conjugated hydrocarbons, Journal of Molecular structure 376 (Issues 1-3) (1996), 495-505. See Table 2 on p. 501.
S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, Unbranched catacondensed polygonal systems containing hexagons and tetragons, Croatica Chem. Acta, 69 (1996), 757-774.
J. Eckhoff, Extremal interval graphs, J. Graph Theory 17 1 (1993), 117-127.
R. M. Foster, Solution to Problem E185, Amer. Math. Monthly, 44 (1937), 50-51.
R. M. Foster, Solution to Problem E185, Amer. Math. Monthly, 44 (1937), 50-51 [annotated scanned copy].
F. Harary and R. W. Robinson, Tapeworms, Unpublished manuscript, circa 1973. (Annotated scanned copy)
Thomas M. Liggett and Wenpin Tang, One-dependent hard-core processes and colorings of the star graph, arXiv:1804.06877 [math.PR], 2018.
L. Markenzon, O. Vernet, and P. R. da Costa Pereira, A clique-difference encoding scheme for labelled k-path graphs, Discrete Appl. Math. 156 (2008), 3216-3222.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Gyula Tasi and Fujio Mizukami, Quantum algebraic-combinatoric study of the conformational properties of n-alkanes, J. Math. Chemistry, 25, 1999, 55-64 (see p. 60).
Eric Weisstein's World of Mathematics, Alkane Graph.
Eric Weisstein's World of Mathematics, Planar Embedding.
Index entries for sequences obtained by enumerating foldings
Index entries for linear recurrences with constant coefficients, signature (4,0,-12,9).

Cf. A036359, A002216, A005963, A000228, A001997, A001444, A038766, A007051.
Column 3 of A320750, offset by one. Column k = 0 of A323942, offset by two.
Cf. A124302 (oriented), A107767 (chiral), A182522 (achiral), with varying offsets.
Column 3 of A320750.
The numbers of unlabeled k-paths for k = 2..7 are given in A005418, A001998, A056323, A056324, A056325, and A345207, respectively.
The sequences above converge to A103293(n+1).

a:=[];; for n in [2..45] do if n mod 2 =0 then Add(a,((3^((n-2)/2)+1)/2)^2); else Add(a,  3^((n-3)/2)+(1/4)*(3^(n-2)+1)); fi; od; a; # Muniru A Asiru, Oct 28 2018

A001998 := proc(n) if n = 0 then 1 elif n mod 2 = 1 then (1/4)*(3^n+4*3^((n-1)/2)+1) else (1/4)*(3^n+2*3^(n/2)+1); fi; end;
A001998:=(-1+3*z+2*z**2-8*z**3+3*z**4)/(z-1)/(3*z-1)/(3*z**2-1); # conjectured by Simon Plouffe in his 1992 dissertation; gives sequence with an extra leading 1

a[n_?OddQ] := (1/4)*(3^n + 4*3^((n - 1)/2) + 1); a[n_?EvenQ] := (1/4)*(3^n + 2*3^(n/2) + 1); Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Jan 25 2013, from formula *)
LinearRecurrence[{4,0,-12,9},{1,2,4,10},30] (* Harvey P. Dale, Apr 10 2013 *)
Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2,k] + Ach[n-2,k-1] + Ach[n-2,k-2]] (* A304972 *)
k=3; Table[Sum[StirlingS2[n,j]+Ach[n,j],{j,k}]/2,{n,40}] (* Robert A. Russell, Oct 28 2018 *)

Vec((1-2*x-4*x^2+6*x^3)/((1-x)*(1-3*x)*(1-3*x^2)) + O(x^50)) \\ Colin Barker, May 15 2016

a(n) = if n mod 2 = 0 then ((3^((n-2)/2)+1)/2)^2 else 3^((n-3)/2)+(1/4)*(3^(n-2)+1).
G.f.: (1-2*x-4*x^2+6*x^3) / ((1-x)*(1-3*x)*(1-3*x^2)). - Corrected by Colin Barker, May 15 2016
a(n) = 4*a(n-1)-12*a(n-3)+9*a(n-4), with a(0)=1, a(1)=2, a(2)=4, a(3)=10. - Harvey P. Dale, Apr 10 2013
a(n) = (1+3^n+3^(1/2*(-1+n))*(2-2*(-1)^n+sqrt(3)+(-1)^n*sqrt(3)))/4. - Colin Barker, May 15 2016
E.g.f.: (2*sqrt(3)*sinh(sqrt(3)*x) + 3*exp(2*x)*cosh(x) + 3*cosh(sqrt(3)*x))/6. - Ilya Gutkovskiy, May 15 2016
From Robert A. Russell, Oct 28 2018: (Start)
a(n-1) = (A124302(n) + A182522(n)) / 2 = A124302(n) - A107767(n-1) = A107767(n-1) + A182522(n).
a(n-1) = Sum_{j=1..k} (S2(n,j) + Ach(n,j)) / 2, where k=3 is the maximum number of colors, S2 is the Stirling subset number A008277, and Ach(n,k) = [n>=0 & n<2 & n==k] + [n>1]*(k*Ach(n-2,k) + Ach(n-2,k-1) + Ach(n-2,k-2)).
a(n-1) = A057427(n) + A056326(n) + A056327(n). (End)
a(2*n) = A007051(n)^2; a(2*n+1) = A007051(n)*A007051(n+1). - Todd Simpson, Mar 25 2024

Offset and Maple code corrected by Colin Mallows, Nov 12 1999

Term added by Robert A. Russell, Oct 30 2018

A019988 Number of ways of embedding a connected graph with n edges in the square lattice.

Original entry on oeis.org

1, 2, 5, 16, 55, 222, 950, 4265, 19591, 91678, 434005, 2073783, 9979772, 48315186, 235088794, 1148891118, 5636168859, 27743309673

Offset: 1

Russ Cox

It is assumed that all edges have length one. - N. J. A. Sloane, Apr 17 2019
These are referred to as 'polysticks', 'polyedges' or 'polyforms'. - Jack W Grahl, Jul 24 2018
Number of connected subgraphs of the square lattice (or grid) containing n length-one line segments. Configurations differing only a rotation or reflection are not counted as different. The question may also be stated in terms of placing unit toothpicks in a connected arrangement on the square lattice. - N. J. A. Sloane, Apr 17 2019
The solution for n=5 features in the card game Digit. - Paweł Rafał Bieliński, Apr 17 2019

Brian R. Barwell, "Polysticks," Journal of Recreational Mathematics, 22 (1990), 165-175.

D. Goodger, An introduction to Polysticks
M. Keller, Counting polyforms
D. Knuth, Dancing Links, arXiv:cs/0011047 [cs.DS], 2000. (A discussion of backtracking algorithms which mentions some problems of polystick tiling.)
Ed Pegg, Jr., Illustrations of polyforms
N. J. A. Sloane, Illustration of a(1)-a(4)
Eric Weisstein's World of Mathematics, Polyedge
Wikicommons, Polysticks 5-sticks 6-sticks 7-sticks

If only translations (but not rotations) are factored, consider fixed polyedges (A096267).
If reflections are considered different, we obtain the one-sided polysticks, counted by (A151537). - Jack W Grahl, Jul 24 2018
Cf. A001997, A003792, A006372, A059103, A085632, A056841 (tree-like), A348095 (with cycles), A348096 (refined by symmetry), A181528.
See A336281 for another version.
6th row of A366766.

A348095(n) + A056841(n+1) = a(n). - R. J. Mathar, Sep 30 2021

More terms from Brendan Owen (brendan_owen(AT)yahoo.com), Feb 20 2002

a(18) from John Mason, Jun 01 2023

A001444 Bending a piece of wire of length n+1 (configurations that can only be brought into coincidence by turning the figure over are counted as different).

Original entry on oeis.org

1, 2, 6, 15, 45, 126, 378, 1107, 3321, 9882, 29646, 88695, 266085, 797526, 2392578, 7175547, 21526641, 64573362, 193720086, 581140575, 1743421725, 5230206126, 15690618378, 47071677987, 141215033961, 423644570442, 1270933711326, 3812799539655, 11438398618965

Offset: 0

Todd Simpson

The wire stays in the plane, there are n bends, each is R,L or O.

			There are 2 ways to bend a piece of wire of length 2 (bend it or not).
G.f. = 1 + 2*x + 6*x^2 + 15*x^3 + 45*x^4 + 126*x^5 + 378*x^6 + ...

Todd Andrew Simpson, "Combinatorial Proofs and Generalizations of Weyl's Denominator Formula", Ph. D. Dissertation, Penn State University, 1994.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for sequences obtained by enumerating foldings
Index entries for linear recurrences with constant coefficients, signature (3,3,-9).

Cf. A001997, A001998.

Cf. A000244.

a001444 n = div (3 ^ n + 3 ^ (div n 2)) 2
-- Reinhard Zumkeller, Jun 30 2013

Maple
```
f := n->(3^floor(n/2)+3^n)/2;
```

CoefficientList[Series[(1-x-3*x^2)/((1-3*x)*(1-3*x^2)),{x,0,30}],x] (* Vincenzo Librandi, Apr 15 2012 *)
LinearRecurrence[{3,3,-9},{1,2,6},40] (* Harvey P. Dale, Dec 30 2012 *)

a(n) = (3^n + 3^floor(n/2))/2.
G.f.: G(0) where G(k) = 1 + x*(3*3^k + 1)*(1 + 3*x*G(k+1))/(1 + 3^k). - Sergei N. Gladkovskii, Dec 13 2011 [Edited by Michael Somos, Sep 09 2013]
E.g.f. E(x) = (exp(3*x)+cosh(x*sqrt(3))+sinh(x*sqrt(3))/sqrt(3))/2 = G(0); G(k) = 1 + x*(3*3^k+1)/((2*k+1)*(1+3^k) - 3*x*(2*k+1)*(1+3^k)/(3*x + (2*k+2)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Dec 13 2011
From Colin Barker, Apr 02 2012: (Start)
a(n) = 3*a(n-1) + 3*a(n-2) - 9*a(n-3).
G.f.: x*(1-x-3*x^2)/((1-3*x)*(1-3*x^2)). (End)

Interpretation in terms of bending wire from Colin Mallows.

A006817 Trails of length n on square lattice.

Original entry on oeis.org

1, 4, 12, 36, 108, 316, 916, 2628, 7500, 21268, 60092, 169092, 474924, 1329188, 3715244, 10359636, 28856252, 80220244, 222847804, 618083972, 1713283628, 4742946484, 13123882524, 36274940740, 100226653420, 276669062116, 763482430316, 2105208491748, 5803285527724

Offset: 0

N. J. A. Sloane

A trail is a path which may cross itself but does not reuse an edge. This sequence counts directed paths on the square lattice.

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

Andrey Zabolotskiy, Table of n, a(n) for n = 0..31 (from Conway & Guttmann; terms 0..30 from Bert Dobbelaere)
A. R. Conway and A. J. Guttmann, Enumeration of self-avoiding trails on a square lattice using a transfer matrix technique, J. Phys. A: Math. Gen., 26 (1993), 1535-1552; arXiv:hep-lat/9211063, 1992.
A. J. Guttmann, Lattice trails II: numerical results, J. Phys. A 22 (1989), 575-588.
A. Malakis, The trail problem on the square lattice, J. Phys A 9 (8) (1976) p 1283. Table 1.

Undirected trails-rotation and reflection are counted by A001997.

Cf. A006818, A006819, A006851.

More terms from David W. Wilson, Jul 20 2001

More terms from Bert Dobbelaere, Jan 19 2019

A066372 Number of different shapes formed by bending a piece of wire of length n in the plane.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 15, 23, 43, 71, 128, 209, 379, 650, 1145, 1928, 3422, 5908, 10295, 17530, 30738, 53088, 91971, 157194, 273621, 471865, 814557, 1393822, 2414895, 4157492, 7160018, 12253782, 21163410, 36381025, 62549316, 107029982, 184430758

Offset: 1

Richard D. Plotz (Dick(AT)Plotz.com), Dec 22 2001

Wire is marked into n equal segments by n-1 marks, is bent at right angles at each of these points, making each segment parallel to one of two rectangular axes. (Stays in plane, bends are of +-90 degs.) May cross itself but is not self-coincident over a finite length. Two configurations which differ only in a rotation or turning over are not counted as different.

In addition to not allowing straight segments, there are two further subtle differences between the counting here and the counting in A001997. In this sequence, if the wire effectively forms a closed loop, then that shape is counted only once, whereas in A001997 the position of the ends of the wire matters. Similarly, the same consideration applies to places where the wire is self-coincident. In this sequence, we assume our eyes are not good enough to distinguish which of two (or more) ways of bending the wire achieve the same shape. These distinctions first matter for n=8 where in this sequence all three arrangements which look like a slanted 8 are equivalent. - Sean A. Irvine, Oct 10 2023

			Let LRUD denote left, right, up, down. Then for n = 1..4 the solutions are R, RD, RDL, RDR, RDLU, RDLD, RDRD.
For n=5 the 5 shapes are:
  __.__.  __.  .__  |__.    .__.     __.
    |__|    |__|       |__| |  |__|    |__.
                                          |__

Deborah Freedman, dlf(AT)alumni.princeton.edu, personal communication.

Erich Friedman, Illustration of initial terms
Ron Knott, Watch Out for Fibonacci Forgeries - Right-Angled Links?
Index entries for sequences obtained by enumerating foldings

See A001997 for another version.

Cf. A046661, A122224 for self-avoiding paths.

a(10)-a(23) from Nathaniel Johnston, Jan 04 2011

a(24)-a(37) from Bert Dobbelaere, Jan 12 2020

A178778 Partial sums of walks of length n+1 on a tetrahedron A001998.

Original entry on oeis.org

1, 3, 7, 17, 42, 112, 308, 882, 2563, 7565, 22449, 66979, 200204, 599514, 1796350, 5385764, 16150725, 48442327, 145307291, 435892341, 1307617966, 3922765316, 11768118792, 35304090646, 105911740487, 317734424289, 953201678533, 2859602644103, 8578803149328

Offset: 0

Jonathan Vos Post, Dec 26 2010

The subsequence of primes begins 3, 7, 17, no more through a(27).

			a(5) = 1 + 2 + 4 + 10 + 25 + 70 = 112.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4,-12,21,-9).

Cf. A001998, A036359, A002216, A005963, A000228, A001997, A001444, A038766.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (6*x^3-4*x^2-2*x+1)/((x-1)^2*(3*x-1)*(3*x^2-1)) )); // G. C. Greubel, Jan 24 2019

CoefficientList[Series[(6*x^3-4*x^2-2*x+1)/((x-1)^2*(3*x-1)*(3*x^2-1)), {x,0,30}], x] (* G. C. Greubel, Jan 24 2019 *)

Vec((1-2*x-4*x^2+6*x^3)/((1-x)^2*(1-3*x)*(1-3*x^2)) + O(x^50)) \\ Colin Barker, May 17 2016

((6*x^3-4*x^2-2*x+1)/((x-1)^2*(3*x-1)*(3*x^2-1))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 24 2019

a(n) = Sum_{i=0..n} (if i mod 2 = 0 then ((3^((i-2)/2)+1)/2)^2 else 3^((i-3)/2)+(1/4)*(3^(i-2)+1)).
G.f.: (6*x^3-4*x^2-2*x+1) / ((x-1)^2*(3*x-1)*(3*x^2-1)). - Colin Barker, Apr 20 2013
From Colin Barker, May 17 2016: (Start)
a(n) = (-7+3^(1+n)+3^(1/2*(-1+n))*(9-9*(-1)^n+5*sqrt(3)+5*(-1)^n*sqrt(3))+2*(1+n))/8.
a(n) = (2*n + 10*3^(n/2) + 3^(n+1) - 5)/8 for n even.
a(n) = (2*n + 3^(n+1) + 2*3^((n+3)/2) - 5)/8 for n odd.
a(n) = 5*a(n-1) - 4*a(n-2) - 12*a(n-3) + 21*a(n-4) - 9*a(n-5) for n>4.
(End)

cp's OEIS Frontend

A000682 Semi-meanders: number of ways a semi-infinite directed curve can cross a straight line n times.

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A001998 Bending a piece of wire of length n+1; walks of length n+1 on a tetrahedron; also non-branched catafusenes with n+2 condensed hexagons.

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A019988 Number of ways of embedding a connected graph with n edges in the square lattice.

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A001444 Bending a piece of wire of length n+1 (configurations that can only be brought into coincidence by turning the figure over are counted as different).

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A006817 Trails of length n on square lattice.

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A066372 Number of different shapes formed by bending a piece of wire of length n in the plane.

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A178778 Partial sums of walks of length n+1 on a tetrahedron A001998.

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