A301620 a(n) is the total number of top arches with exactly one covering arch for semi-meanders with n top arches.

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

Author

Roger Ford, Mar 24 2018

Keywords

• nonn

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

Examples

			For n = 4, a(4) = 4.  + + are underneath the starting and ending of each arch with exactly one covering arch.
          /\                  /\
         //\\         /\     //\\       /\
      /\///\\\,  /\/\//\\,  ///\\\/\,  //\\/\/\ .
         +  +         ++     +  +       ++
		

Links

• Jean-François Alcover, Table of n, a(n) for n = 1..43

Crossrefs

Cf. A000682, A259689.

Programs

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 *)

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