%I
%S 1,2,4,6,14,18,46,54,146,162,454,486,1394,1458,4246,4374,12866,13122,
%T 38854,39366,117074,118098,352246,354294,1058786,1062882,3180454,
%U 3188646,9549554,9565938,28665046,28697814,86027906,86093442,258149254
%N Number of permutations of {1,2,3,...,n} where, for 1 < i <= n, the ith number has maximized sum of the i1 absolute differences from all previous numbers of the permutation.
%C Another variant of A095236: Here each phone after the first selected (which can still be any) is chosen such that the total distance in the normal sense from the chosen phone to all previouslychosen phones in the row is maximized. (Equivalently, the average distance is maximized.) Another space or privacyconscious selection strategy. Are there any applications of this sequence to phyllotaxy? Gregarious (or eavesdropping) strategy: If, instead, the total (average) distance is minimized, the sequence generated is 1,2,4,8,16,32,64,128,256,512,..., apparently the nonnegative powers of 2.
%C In the gregarious case (suggested by the above comment), the permutations that result are exactly those that avoid the permutation patterns 132 and 312. See link to Art of Problem Solving Forums for proof of formula below.  _Joel B. Lewis_, May 16 2009
%H Problem solved on the Art of Problem Solving forum, <a href="http://www.artofproblemsolving.com/Forum/viewtopic.php?t=277009">Urinalchoice permutations</a>. [From _Joel B. Lewis_, May 16 2009]
%F a(1) = 1; Conjectured: For k >= 1, a(2k) = a(2k1) + 2^(k1) and a(2k+1) = 2*a(2k1) + a(2k) (needs proof or a reference).
%F a(2n) = 2 * 3^(n  1) for n >= 1. a(2n + 1) = 2 * 3^n  2^n for n >= 0.  _Joel B. Lewis_, May 16 2009
%F Conjecture: a(n) = 5*a(n2)6*a(n4). G.f.: x*(1+2*xx^24*x^3)/((12*x^2)*(13*x^2)).  _Colin Barker_, Jul 27 2012
%F Conjecture: a(n) = 2^(((1)^n + 2*n5)/4)*((1)^n1)  2*3^(((1)^n + 2*n5)/4)*((1)^n2).  _Luce ETIENNE_, Dec 20 2014
%e a(4)=6 as these six permutations of {1,2,3,4} are counted (as in A095236(4)): (1,4,2,3), (1,4,3,2), (2,4,1,3), (3,1,4,2), (4,1,2,3) and (4,1,3,2).
%e In particular, (2,4,3,1) and (3,1,2,4), counted in A095236(4), are not counted here.
%Y Cf. A095236.
%Y Taking every other term gives A008776 (evenindexed terms) and A027649 (oddindexed terms).  _Joel B. Lewis_, May 16 2009
%K nonn
%O 1,2
%A _Rick L. Shepherd_, Jul 06 2004
%E More terms from _Joel B. Lewis_, May 16 2009
