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
Examples
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
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..40
- R. J. Mathar, ASCII representations
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
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