

A095698


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.


0



1, 2, 4, 6, 14, 18, 46, 54, 146, 162, 454, 486, 1394, 1458, 4246, 4374, 12866, 13122, 38854, 39366, 117074, 118098, 352246, 354294, 1058786, 1062882, 3180454, 3188646, 9549554, 9565938, 28665046, 28697814, 86027906, 86093442, 258149254
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

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.
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


LINKS

Table of n, a(n) for n=1..35.
Problem solved on the Art of Problem Solving forum, Urinalchoice permutations. [From Joel B. Lewis, May 16 2009]


FORMULA

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).
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
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
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


EXAMPLE

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).
In particular, (2,4,3,1) and (3,1,2,4), counted in A095236(4), are not counted here.


CROSSREFS

Cf. A095236.
Taking every other term gives A008776 (evenindexed terms) and A027649 (oddindexed terms).  Joel B. Lewis, May 16 2009
Sequence in context: A138307 A323101 A124693 * A277909 A064409 A225078
Adjacent sequences: A095695 A095696 A095697 * A095699 A095700 A095701


KEYWORD

nonn


AUTHOR

Rick L. Shepherd, Jul 06 2004


EXTENSIONS

More terms from Joel B. Lewis, May 16 2009


STATUS

approved



