0,3

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 (mathar(AT)strw.leidenuniv.nl), May 06 2006

R. J. Mathar ASCII representation n<=11.

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

Cf. A005316, A076875, A076906, A076907.

Sequence in context: A007165 A107321 A005316 this_sequence A124495 A007919 A069752

Adjacent sequences: A076873 A076874 A076875 this_sequence A076877 A076878 A076879

nonn,more,nice

N. J. A. Sloane (njas(AT)research.att.com) and Jon Wild (wild(AT)music.mcgill.ca), Nov 26, 2002

More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 04 2007