A077014 - OEIS

A077014

Number of ways that a directed line (or river) that starts in the south can cross an east-west road n times.

2, 1, 2, 2, 6, 8, 28, 42, 162, 262, 1076, 1828, 7852, 13820, 61388, 110954, 505878, 933458, 4345660, 8152860, 38608380, 73424650, 352686780, 678390116, 3298016912, 6405031050, 31461151108, 61606881612, 305327366988, 602188541928, 3007925909860, 5969806669034

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

The difference between this and A005316 is that here the river can also end up in the southwest. In A005316 it was required to end up in the northeast or southeast.

Or, number of ways that an undirected line that starts and ends anywhere can cross an east-west road n times.

LINKS

Hunter Hogan, Table of n, a(n) for n = 0..55

Hunter Hogan, mapFolding: Create, alter, and optimize algorithms to count meanders, stamp foldings, and map foldings, GitHub repository, (2026).

Stéphane Legendre, Foldings and Meanders, arXiv preprint arXiv:1302.2025 [math.CO], 2013.

Stéphane Legendre, Foldings and Meanders, Australasian Journal of Combinatorics 58(2) (2014), 275-291.

FORMULA

Equals A005316(n) if n odd, or 2*A005316(n) if n even (because then each solution to A005316 can be reflected in a vertical mirror to give another solution).

From Hunter Hogan, Aug 06 2026: (Start)

a(0) = 2, a(n) = A000682(n+1) - A223093(n).

a(0) = 2, a(n) = A223095(n) - A000136(n) + 2*A000682(n+1). (End)

MATHEMATICA

A005316 = Cases[Import["https://oeis.org/A005316/b005316.txt", "Table"], {_, _}][[All, 2]];

a[n_] := If[OddQ[n], A005316[[n+1]], 2*A005316[[n+1]] ];

a /@ Range[0, 31] (* Jean-François Alcover, Sep 07 2019 *)

CROSSREFS

Cf. A005316.

Cf. A000682 (semi-meanders), A223093 (foldings of labeled stamps with opposite orientation end leaves).

Cf. A223095 (foldings of labeled stamps with inward end leaves), A000136 (foldings of labeled stamps).

Sequence in context: A081727 A000020 A396907 * A093655 A023140 A145859