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A209350 Number of initially rising meander words, where each letter of the cyclic n-ary alphabet occurs twice.

%I A209350 #18 Jan 02 2020 14:33:06
%S A209350 1,0,1,5,9,11,16,19,25,29,36,41,49,55,64,71,81,89,100,109,121,131,144,
%T A209350 155,169,181,196,209,225,239,256,271,289,305,324,341,361,379,400,419,
%U A209350 441,461,484,505,529,551,576,599,625,649,676,701,729,755,784,811,841
%N A209350 Number of initially rising meander words, where each letter of the cyclic n-ary alphabet occurs twice.
%C A209350 In a meander word letters of neighboring positions have to be neighbors in the alphabet, where in a cyclic alphabet the first and the last letters are considered neighbors too.  The words are not considered cyclic here.
%C A209350 A word is initially rising if it is empty or if it begins with the first letter of the alphabet that can only be followed by the second letter in this word position.
%C A209350 a(n) is also the number of (2*n-1)-step walks on n-dimensional cubic lattice from (1,0,...,0) to (2,2,...,2) with positive unit steps in all dimensions such that the indices of dimensions used in consecutive steps differ by 1 or are in the set {1,n}.
%H A209350 Alois P. Heinz, <a href="/A209350/b209350.txt">Table of n, a(n) for n = 0..1000</a>
%H A209350 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F A209350 G.f.: -(3*x^6-5*x^5-2*x^4+5*x^3+x^2-2*x+1) / ((x+1)*(x-1)^3).
%F A209350 a(n) = (n-1)^2 if n<3, a(n) = (n/2+1)^2 - (n mod 2)*5/4 else.
%e A209350 a(0) = 1: the empty word.
%e A209350 a(1) = 0 = |{ }|.
%e A209350 a(2) = 1 = |{abab}|.
%e A209350 a(3) = 5 = |{abacbc, abcabc, abcacb, abcbac, abcbca}|.
%e A209350 a(4) = 9 = |{ababcdcd, abadcbcd, abadcdcb, abcbadcd, abcbcdad, abcdabcd, abcdadcb, abcdcbad, abcdcdab}|.
%p A209350 a:= n-> `if`(n<3, (n-1)^2, (n/2+1)^2 -(n mod 2)*5/4):
%p A209350 seq(a(n), n=0..60);
%t A209350 LinearRecurrence[{2,0,-2,1},{1,0,1,5,9,11,16},60] (* _Harvey P. Dale_, Jan 02 2020 *)
%Y A209350 Row n=2 of A209349.
%Y A209350 First differences for n>2 give: A084964(n+1), A097065(n+3).
%Y A209350 Cf. A245578.
%K A209350 nonn,walk,easy
%O A209350 0,4
%A A209350 _Alois P. Heinz_, Mar 06 2012