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Encyclopedia > Meandric number

In mathematics, a meander or closed meander is a self-avoiding closed curve which intersects a line a number of times. Intuitively, a meander can be viewed as a road crossing a river through a number of bridges.

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Given a fixed oriented line L in the Euclidean plane R2, a meander of order n is a non-self-intersecting closed curve in R2 which transversally intersects the line at 2n points for some positive integer n. Two meanders are said to be equivalent if they are homeomorphic in the plane.

### Examples

The meander of order 1 intersects the line twice: The meanders of order 2 intersect the line four times:  ### Meandric numbers

The number of distinct meanders of order n is the meandric number Mn. The first fifteen meandric numbers are given below (sequence A005315 in OEIS).

M1 = 1
M2 = 2
M3 = 8
M4 = 42
M5 = 262
M6 = 1828
M7 = 13820
M8 = 110954
M9 = 933458
M10 = 8152860
M11 = 73424650
M12 = 678390116
M13 = 6405031050
M14 = 61606881612
M15 = 602188541928

## Open meander

Given a fixed oriented line L in the Euclidean plane R2, an open meander of order n is a non-self-intersecting oriented curve in R2 which transversally intersects the line at n points for some positive integer n. Two open meanders are said to be equivalent if they are homeomorphic in the plane.

### Examples

The open meander of order 1 intersects the line once: The open meander of order 2 intersects the line twice: ### Open meandric numbers

The number of distinct open meanders of order n is the open meandric number mn. The first fifteen open meandric numbers are given below (sequence A005316 in OEIS).

m1 = 1
m2 = 1
m3 = 2
m4 = 3
m5 = 8
m6 = 14
m7 = 42
m8 = 81
m9 = 262
m10 = 538
m11 = 1828
m12 = 3926
m13 = 13820
m14 = 30694
m15 = 110954

## Semi-meander

Given a fixed oriented ray R in the Euclidean plane R2, a semi-meander of order n is a non-self-intersecting closed curve in R2 which transversally intersects the ray at n points for some positive integer n. Two semi-meanders are said to be equivalent if they are homeomorphic in the plane.

### Examples

The semi-meander of order 1 intersects the ray once: The semi-meander of order 2 intersects the ray twice: ### Semi-meandric numbers

The number of distinct semi-meanders of order n is the semi-meandric number Mn (usually denoted with an overline instead of an underline). The first fifteen semi-meandric numbers are given below (sequence A000682 in OEIS).

M1 = 1
M2 = 1
M3 = 2
M4 = 4
M5 = 10
M6 = 24
M7 = 66
M8 = 174
M9 = 504
M10 = 1406
M11 = 4210
M12 = 12198
M13 = 37378
M14 = 111278
M15 = 346846

## Properties of meandric numbers

There is an injective function from meandric to open meandric numbers:

Mn = m2n−1

Each meandric number can be bounded by semi-meandric numbers:

MnMnM2n

For n > 1, meandric numbers are even:

Mn ≡ 0 (mod 2) Results from FactBites:

 42 (number) - Wikipedia, the free encyclopedia (954 words) It is also the third 15-gonal number, a Catalan number, a meandric number, an open meandric number, a Harshad number and a self number. The movement number of the Hallelujah Chorus in Handel's Messiah. The number of perfect squares formed by the grid at a 19x19 Go board, when the "squares" are slightly rectangular with the ratio 13/12, as required by tradition.
 2 (number) - Wikipedia, the free encyclopedia (1204 words) Taking the square root of a number is such a common mathematical operation, that the spot on the root sign where the exponent would normally be written for cubic roots and other such roots, is left blank for square roots, as it is considered tacit. Two is also a Motzkin number, a Bell number, an all-Harshad number, a meandric number, a semi-meandric number, and an open meandric number. In rugby union, 2 is the number of the hooker.
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