A076876 Meandric numbers for a river crossing two parallel roads at n points.

1, 1, 2, 3, 8, 14, 43, 81, 272, 538, 1920, 3926, 14649, 30694, 118489, 252939, 1002994, 2172830, 8805410, 19304190, 79648888, 176343390, 738665040, 1649008456, 6996865599, 15730575554, 67491558466, 152663683494, 661370687363, 1503962954930, 6571177867129

Offset

0,3

Comments

a(n) = number of ways that a curve can start in the (-,-) quadrant, cross two parallel lines and end up in the (+,+) or (+,-) quadrant if n is even or head East between the two roads if n is odd.

A107321 is a lower bound. - R. J. Mathar, May 06 2006

It appears that for odd n, A076876(n) = A005316(n+1). And for even n, A076876(n) >= A005316(n+1). If this is the case then a(21)=176343390, a(23)=1649008456, etc. - Robert Price, Jul 27 2013.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..40

R. J. Mathar, ASCII representations

Example

Let b(n) = A005316(n). Then a(0) = b(0), a(1) = b(1), a(2) = b(1) + b(2), a(3) = b(3) + b(2), a(4) = b(4) + 2*b(3) + 1, a(5) = b(5) + b(3)*b(2) + b(4) + 1.
Consider n=5: if we do not cross the second road there are b(5) = 8 solutions. If we cross the first road 3 times and then the second road twice there are b(3)*b(2) = 2 solutions. If we cross the first road once and the second road 4 times there are b(4) = 3 solutions. The only other possibility is to cross road 1, road 2 twice, road 1 twice and exit to the right.
For larger n it is convenient to give the vector of the number of times the same road is crossed. For example for n=6 the vectors and the numbers of possibilities are as follows:
[6] ...... 14
[5 1] ..... 8
[3 3] ..... 4
[3 2 1] ... 2
[1 5] ..... 8
[1 4 1] ... 3
[1 2 3] ... 2
[1 2 2 1] . 2
Total .... 43

Crossrefs

Cf. A005316, A206432, A204352, A076875, A076906, A076907.

Sequence in context: A007165 A107321 A005316 * A124495 A007919 A249167

Adjacent sequences: A076873 A076874 A076875 * A076877 A076878 A076879

Keyword

nonn,nice

Author

N. J. A. Sloane and Jon Wild, Nov 26 2002

Extensions

More terms from R. J. Mathar, Mar 04 2007

a(12)-a(20) from Robert Price, Apr 15 2012

a(21)-a(40) from Andrew Howroyd, Dec 07 2015

Status

approved