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

%I A076876 #35 Apr 18 2025 23:33:30
%S A076876 1,1,2,3,8,14,43,81,272,538,1920,3926,14649,30694,118489,252939,
%T A076876 1002994,2172830,8805410,19304190,79648888,176343390,738665040,
%U A076876 1649008456,6996865599,15730575554,67491558466,152663683494,661370687363,1503962954930,6571177867129
%N A076876 Meandric numbers for a river crossing two parallel roads at n points.
%C A076876 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.
%C A076876 A107321 is a lower bound. - _R. J. Mathar_, May 06 2006
%C A076876 It appears that for odd n, A076876(n) = A005316(n+1). And for even n, A076876(n) >= A005316(n+1). - _Robert Price_, Jul 27 2013.
%H A076876 Andrew Howroyd, <a href="/A076876/b076876.txt">Table of n, a(n) for n = 0..40</a>
%H A076876 R. J. Mathar, <a href="/A076876/a076876.txt">ASCII representations</a>
%e A076876 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.
%e A076876 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.
%e A076876 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:
%e A076876 [6] ...... 14
%e A076876 [5 1] ..... 8
%e A076876 [3 3] ..... 4
%e A076876 [3 2 1] ... 2
%e A076876 [1 5] ..... 8
%e A076876 [1 4 1] ... 3
%e A076876 [1 2 3] ... 2
%e A076876 [1 2 2 1] . 2
%e A076876 Total .... 43
%Y A076876 Column k=2 of A380367.
%Y A076876 Cf. A005316, A206432, A204352, A076875, A076906, A076907.
%K A076876 nonn,nice
%O A076876 0,3
%A A076876 _N. J. A. Sloane_ and _Jon Wild_, Nov 26 2002
%E A076876 More terms from _R. J. Mathar_, Mar 04 2007
%E A076876 a(12)-a(20) from _Robert Price_, Apr 15 2012
%E A076876 a(21)-a(40) from _Andrew Howroyd_, Dec 07 2015