1,2

Equivalently, walks on triangle, visiting n+2 vertices, so length n+1, n "corners"; the symmetry group is S3, reversing a walk does not count as different. Walks are not self-avoiding.

Slavik V. Jablan (jablans(AT)yahoo.com) observes that this is also the number of rational knots and links with n crossings (cf. A018240). See reference.

Number of bit strings of length n, not counting strings which are the end-for-end reversal or the 0-for-1 reversal of each other as different. - Carl Witty (cwitty(AT)newtonlabs.com), Oct 27 2001

The formula given in page 1095 of the Balasubramanian reference can be used to derive this sequence. - Parthasarathy Nambi (PachaNambi(AT)yahoo.com), May 14 2007

a(n) is union A007582(n-1) and A161168(n-1). a(n) is union A007582(n-1) and A063376(n-1). a(n) written in base 2 (see A164370): a(1) = 1, a(2) = 10, a(3) = 11, a(n) for n >= 4: 1010,10100,100100,1001000,10001000,100010000,1000010000, ..., i.e. digit 1, (A004526(n-3)) times 0, digit 1, (A004526(n-2)) times 0. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Aug 14 2009]

Also number of compositions of n up to direction, where a composition is considered equivalent to its reversal. [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Oct 24 2009]

K. Balasubramanian, "Combinatorial Enumeration of Chemical Isomers", Indian J. Chem., (1978) vol. 16B, pp. 1094-1096. See page 1095.

S. J. Cyvin et al., Theory of polypentagons, J. Chem. Inf. Comput. Sci., 33 (1993), 466-474.

N. Hoffman, Binary grids and a related counting problem, 2-Year Coll. Math. J., 9 (1978), 267-272.

Jablan S. and Sazdanovic R., LinKnot: Knot Theory by Computer, World Scientific Press, 2007.

Joseph S. Madachy: Madachy's Mathematical Recreations. New York: Dover Publications, Inc., 1979, p. 46 (first publ. by Charles Scribner's Sons, New York, 1966, under the title: Mathematics on Vacation)

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

Joerg Arndt, Fxtbook

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. V. Jablan, Geometry of Links, XII Yugoslav Geometric Seminar (Novi Sad, 1998), Novi Sad J. Math. 29 (1999), no. 3, 121-139.

N. J. A. Sloane, Classic Sequences

R. A. Sulanke, Moments of generalized Motzkin paths, J. Integer Sequences, Vol. 3 (2000), #00.1.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Barker Code

Eric Weisstein's World of Mathematics, Losanitsch's Triangle

2^(n-2) + 2^([n/2]-1).

G.f.: x*(1+2*x)*(1-3*x^2)/((1-4*x^2)*(1-2*x^2)). - Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), May 08 2001

a(n) = 6a(n-2)-8a(n-4). a(2n) = A063376(n-1) = 2*a(2n-1); a(2n+1) = A007582(n). - Henry Bottomley (se16(AT)btinternet.com), Jul 14 2001

a(n+2) = 2*a(n+1) - A077957(n) with a(1)=1, a(2)=2 [From Yosu Yurramendi (yosu.yurramendi(AT)ehu.es), Oct 24 2008]

a(n)=2a(n-1)+2a(n-2)-4a(n-3) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Dec 05 2008]

A005418 := n->2^(n-2)+2^(floor(n/2)-1);

A005418:=-(-1+3*z**2)/(2*z-1)/(2*z**2-1); [Conjectured by S. Plouffe in his 1992 dissertation.]

Cf. A001998, A001444, A051436, A051437, A007582, A001445.

Cf. A141447 [From Yosu Yurramendi (yosu.yurramendi(AT)ehu.es), Oct 24 2008]

Sequence in context: A052525 A006606 A120421 this_sequence A002215 A007562 A008929

Adjacent sequences: A005415 A005416 A005417 this_sequence A005419 A005420 A005421

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com), R. K. Guy

Interpretation in terms of walks from Colin Mallows (colinm(AT)research.avayalabs.com).