

A076876


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


24



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
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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



