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A260785 Leading diagonal of triangle in A259689: a(n)= number of permutations without overlaps that generate exactly 2 permutations without overlaps in a(n+1).

%I A260785 #27 Jan 28 2019 13:57:52
%S A260785 1,2,2,6,10,32,68,220,528,1724,4460,14664,39908,131944,372448,1237016,
%T A260785 3589384,11967140,35479312,118675768,357957984,1200724776,3673173656,
%U A260785 12351611656,38232022416
%N A260785 Leading diagonal of triangle in A259689: a(n)= number of permutations without overlaps that generate exactly 2 permutations without overlaps in a(n+1).
%C A260785 From _Roger Ford_, Oct 12 2015: (Start)
%C A260785 a(n)= Number of semi-meander solutions for n with 2 returns to the x axis (or number of 2 distinct arch groups).
%C A260785 Example:  n=5  -= return to x axis
%C A260785       /\             /\                        /\
%C A260785      //\\           /  \             /\       //\\
%C A260785     ///\\\         /  /\\      /\   //\\     ///\\\
%C A260785 /\-////\\\\-   /\-//\//\\\-   //\\-///\\\-  ////\\\\-/\-
%C A260785      /\
%C A260785     /  \           /\
%C A260785    //\  \         //\\   /\
%C A260785   ///\\/\\-/\-   ///\\\-//\\-      a(5)=6.
%C A260785 a(n)= Number of hills (arches with a peak at 1 and no covering arches) for semi-meander solutions with n-1 arches.
%C A260785 Example:  n=5   semi-meander solutions with 4 arches (/\)= hill
%C A260785                               /\              /\
%C A260785          /\    /\            //\\            //\\
%C A260785 (/\)(/\)//\\  //\\(/\)(/\)  ///\\\(/\)  (/\)///\\\  a(5)=6.
%C A260785 (End)
%C A260785 From _Roger Ford_, Jan 27 2018: (Start)
%C A260785 a(n)= Number of solutions for folding a strip of n stamps with stamp 1 on top and each solution ordering having the absolute value of the difference of the stamp number before and after stamp n equal to 1. (If stamp n is the last stamp in the solution ordering then add a(1) to the end of the ordering.)
%C A260785 Example:  n=5
%C A260785 12354  |3-4| = 1,  14325(1)  |2-1| = 1,  12453  |4-3| = 1,
%C A260785 14532  |4-3| = 1,  15234  |1-2| = 1,  13542  |3-4| = 1,  a(5)=6.
%C A260785 (End)
%H A260785 Albert Sade, <a href="/A000108/a000108_17.pdf">Sur les Chevauchements des Permutations</a>, published by the author, Marseille, 1949. [Annotated scanned copy]
%Y A260785 Column k=2 of A259689.
%K A260785 nonn,more
%O A260785 2,2
%A A260785 _N. J. A. Sloane_, Aug 04 2015
%E A260785 Corrected and extended by _Roger Ford_, Oct 12 2015
%E A260785 a(14)-(26) from _Andrew Howroyd_, Dec 05 2018