cp's OEIS Frontend

A095698 Number of permutations of {1,2,3,...,n} where, for 1 < i <= n, the i-th number has maximized sum of the i-1 absolute differences from all previous numbers of the permutation.

%I A095698 #33 Nov 19 2022 20:26:21
%S A095698 1,2,4,6,14,18,46,54,146,162,454,486,1394,1458,4246,4374,12866,13122,
%T A095698 38854,39366,117074,118098,352246,354294,1058786,1062882,3180454,
%U A095698 3188646,9549554,9565938,28665046,28697814,86027906,86093442,258149254
%N A095698 Number of permutations of {1,2,3,...,n} where, for 1 < i <= n, the i-th number has maximized sum of the i-1 absolute differences from all previous numbers of the permutation.
%C A095698 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 previously-chosen phones in the row is maximized. (Equivalently, the average distance is maximized.) Another space- or privacy-conscious 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 A095698 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
%C A095698 Taking every other term gives A008776 (even-indexed terms) and A027649 (odd-indexed terms). - _Joel B. Lewis_, May 16 2009
%C A095698 With Lewis's formulas, the addition of the g.f.s for a(2*n) and a(2*n+1) yields the conjectures below: 2*x/(-3*x^2+1) - (-x^2+1)/(-6*x^4+5*x^2-1) = (-4*x^3-x^2+2*x+1)/(6*x^4-5*x^2+1). - _Georg Fischer_, Nov 19 2022
%H A095698 Problem solved on the Art of Problem Solving forum, <a href="https://artofproblemsolving.com/community/c6h277009p1498936">Urinal-choice permutations</a>. [From _Joel B. Lewis_, May 16 2009, corrected from pers. comm.]
%H A095698 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0, 5, 0, -6).
%F A095698 a(1) = 1; Conjectured: For k >= 1, a(2k) = a(2k-1) + 2^(k-1) and a(2k+1) = 2*a(2k-1) + a(2k) (needs proof or a reference).
%F A095698 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 A095698 Conjecture: a(n) = 5*a(n-2) - 6*a(n-4); g.f.: x*(1+2*x-x^2-4*x^3)/((1-2*x^2)*(1-3*x^2)). - _Colin Barker_, Jul 27 2012
%F A095698 Conjecture: a(n) = 2^(((-1)^n + 2*n - 5)/4)*((-1)^n-1) - 2*3^(((-1)^n + 2*n - 5)/4)*((-1)^n-2). - _Luce ETIENNE_, Dec 20 2014
%e A095698 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 A095698 In particular, (2,4,3,1) and (3,1,2,4), counted in A095236(4), are not counted here.
%t A095698 CoefficientList[Series[(-4*x^3-x^2+2*x+1)/(6*x^4-5*x^2+1), {x,0,34}], x] (* _Georg Fischer_, Nov 19 2022 *)
%Y A095698 Cf. A008776, A027649, A095236.
%K A095698 nonn,easy
%O A095698 1,2
%A A095698 _Rick L. Shepherd_, Jul 06 2004
%E A095698 More terms from _Joel B. Lewis_, May 16 2009