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A301620
a(n) is the total number of top arches with exactly one covering arch for semi-meanders with n top arches.
5
0, 0, 2, 4, 18, 42, 156, 398, 1398, 3778, 12982, 36522, 124290, 360182, 1220440, 3618090, 12237698, 36938158, 124880222, 382471606, 1293363816, 4009185912, 13565790984, 42478788432, 143851766298, 454339269482, 1539997455570, 4900091676662, 16624834778474, 53240459608298
OFFSET
1,3
COMMENTS
For n>2, a(n-2) is the number of ways to fold a strip of n stamps with leaf 1 on top and the n leaf not adjacent to the n-1 leaf. Example n = 6, a(6-2) = 4: 125436, 126345, 154362, 163452. - Roger Ford, Mar 29 2019
For n>2, a(n-2) is the number of ways to fold a strip of n stamps with leaf 1 on top and leaf 2 not in the second position and not in the n-th position. Example, for n = 6, a(6-2) = 4: 143265, 156234, 165234, 143256. - Roger Ford, Mar 12 2021
LINKS
Jean-François Alcover, Table of n, a(n) for n = 1..43
FORMULA
a(n) = A000682(n+2) - 2*A000682(n+1).
a(n) = Sum_{k=3..floor((n+3)/2)} (A259689(n+1,k)*(k-2)). - Roger Ford, Dec 10 2018
a(n) = 2*A259702(n+2). - Roger Ford, Dec 24 2018
EXAMPLE
For n = 4, a(4) = 4. + + are underneath the starting and ending of each arch with exactly one covering arch.
/\ /\
//\\ /\ //\\ /\
/\///\\\, /\/\//\\, ///\\\/\, //\\/\/\ .
+ + ++ + + ++
MATHEMATICA
A000682 = Import["https://oeis.org/A000682/b000682.txt", "Table"][[All, 2]];
a[n_] := A000682[[n + 2]] - 2*A000682[[n + 1]];
Array[a, 30] (* Jean-François Alcover, Sep 02 2019 *)
CROSSREFS
Sequence in context: A343529 A143533 A064723 * A240316 A151449 A045664
KEYWORD
nonn
AUTHOR
Roger Ford, Mar 24 2018
STATUS
approved